Nonmaximally-entangled-state quantum photolithography

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[精彩]量子力学英语词汇

1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equati on 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。

On the negative effective mass density in acoustic metamaterials

There is no actual negative mass/mass density in real materials indeed. The negative effective mass/mass density is the result of inaccurate modeling of acoustic metamaterials. In the present study, we consider a one-dimensional lattice consisting of lattice masses each of which contains an internal mass. This simple lattice system is employed to illustrate how the
On the negative effective mass density in acoustic metamaterials
H.H. Huang a, C.T. Sun a,*,G.L. Huang b
a School of Aeronautics and Astronautics, Purdue University, W. Lafayette, IN 47907, USA b Department of Systems Engineering, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA
* Corresponding author. Tel.: +1 765 494 5130; fax: +1 765 494 0307. E-mail address: sun@ (C.T. Sun).

Quantum entanglement

Quantum entanglementMaciej LewensteinMaciej Lewenstein has obtained his degree in Physics from Warsaw University. From 1980 he worked at the Center for Theoretical Physics of the Polish Academy of Sciences. He received his doctoral degree in 1983 at the University of Essen and habilitation in 1986 in Warsaw. He became a full Professor in Poland in 1993. In 1995 he joined “Service de Photones, Atomes et Molecules” of CEA in Saclay. In 1998 he became a full professor and a head of the quantum optics theory group at the University of Hannover. In 2005 he started a new theory group at the “Insitut de Ciencias Fotoniques” in Barcelona. His research interests include: quantum optics, quantum information and statistical physics.Chiara MacchiavelloChiara Macchiavello finished her degree in Physics in 1991 and her PhD in 1995 at the University of Pavia. She held a post-doctoral for two years at the University of Oxford. Since 1998 she has been an Assistant Professor at the University of Pavia.Her research interests include quantum information processing and quantum optics.Dagmar BrussSince 2003 Dagmar Bruss is a professor at the Institute of Theoretical Physics at the University of Duesseldorf, Germany. Her research interests include the foundations of quantum information theory, classification of entanglement and quantum optical implementations of quantum computation.AbstractEntanglement is a fundamental resource in quantum information theory. It allows performing new kinds of communication, such as quantum teleportation and quantum dense coding. It is an essential ingredient in some quantum cryptographic protocols and in quantum algorithms. We give a brief overview of the concept of entanglement in quantum mechanics, and discuss the major results and open problems related to the recent scientific progress in this field.IntroductionEntanglement is a property of the states of quantum systems that are composed of many parties, nowadays frequently called Alice, Bob, Charles etc. Entanglement expresses particularly strong correlations between these parties, persistent even in the case of large separations among the parties, and going beyond simple intuition.Historically, the concept of entanglement goes back to the famous Einstein-Podolski-Rosen (EPR) “paradox”. Einstein, who discovered relativity theory and the modern meaning of causality, was never really happy with quantum mechanics. In his opinion every reasonable physical theory should exhibit a so called local realism.Suppose that we consider two particles, one of which is sent to Alice and one to Bob, and we perform independent local measurements of “reasonable” physical observables on these particles. Of course, the results might be correlated, because the particles come from the same source. But Einstein wanted really to restrict the correlations for “reasonable” physical observables to the ones that result from statistical distributions of some hidden (i.e. unknown to us and not controlled by us) variables that characterize the source of the particles. Since quantum mechanics did not seem to produce correlations consistent with a local hidden variable (LHV) model, Einstein concluded that quantum mechanics is not a complete theory. Erwin Schrödinger, in answer to Einstein’s doubts, introduced in 1935 the term “Verschränkung” (in English “entanglement”) in order to describe these particularly strong quantum mechanical correlations.Entanglement was since then a subject of intense discussions among experts in the foundations of quantum mechanics and philosophers of science (and not only science). It took, however, nearly 30 years until John Bell was able to set the framework for experimental investigations on the question of local realism. Bell formulated his famous inequalities, which have to be fulfilled in any multiparty system described by a LHV model. Alain Aspect and coworkers in Paris have demonstrated in their seminal experiment in 1981 that quantum mechanical states violate these inequalities. Recent very precise experiments of Anton Zeilinger’s group in Vienna confirmed fully Aspect’s demonstrations. All these experiments indicate the correctness of quantum mechanics, and despite various loopholes, they exclude the possibility of LHV models describing properly the physics of the considered systems.Entanglement has become again the subject of cover pages news in the 90’s, when quantum information was born. It was very quickly realized that entanglement is one of the most important resources for quantum information processing. Entanglement is a necessary ingredient for quantum cryptography, quantum teleportation, quantum densecoding, and if not necessary, then at least a much desired ingredient for quantum computing.At the same time the theory of entanglement is related to some of the open questions of mathematics, or more precisely linear algebra and functional analysis. A solution of the entanglement problem could help to characterize the so called positive linear maps, i.e. linear transformations of positive definite operators (or physically speaking quantum mechanical density matrices, see below) into positive definite operators.Entanglement of pure statesIn quantum mechanics (QM) a state of a quantum system corresponds to a vector |Psi> in some vector space, called Hilbert space. Such states are called pure states. One of the most important properties of QM is that linear superpositions of state-vectors are also legitimate state-vectors. This superposition principle lies at the heart of the matter-wave dualism and of quantum interference phenomena.Entanglement is also a result of superposition, but in the composite space of the involved parties. Let us for the moment focus on two parties, Alice and Bob. It is then easy to define states which are not entangled. Such states are product states of the form |Φ>= |a>|b>, i.e. Alice has at her disposal |a>, while Bob has |b>. Product states obviously carry no correlations between Alice and Bob. Entangled pure states may be now defined as those which are superpositions of at least two product states, such as|Φ> = α1|a1>|b1> + α2|a2>|b2> + etc.but cannot be written as a single product state in any other basis. All entangled pure states contain strong quantum mechanical correlations, and do not admit LHV models.Entanglement of mixed states and the separability problemVerify whether a given state-vector is a product state or not is a relatively easy task. In practice, however, we often either do not have full information about the system, or are not able to prepare a desired state perfectly. In effect in everyday situations we deal practically always with statistical mixtures of pure states. There exists a very convenient way to represent such mixtures as so called density operators, or matrices. A density matrix rho corresponding to a pure state-vector |Φ> is a projector onto this state. More general density matrices can be represented as sums of projectors onto pure state-vectors weighted by the corresponding probabilities.The definition of entangled mixed states for composite systems has been formulated by Reinhard Werner from Braunschweig in 1989. In fact, this definition determines which states are not entangled. Non-entangled states, called separable states, are mixtures of pure product states, i.e. convex sums of projectors onto product vectors:ρ = Σι pi|ai>|bi><ai|<bi|, (*)where 0 ≤ pi ≤ 1 are probabilities, i.e. Σιpi= 1. The physical interpretation of thisdefinition is simple: a separable state can be prepared by Alice and Bob by using local operations and classical communication. Checking whether a given state is separable or not is a notoriously difficult task, since one has to check whether the decomposition (*) exists or not. This difficult problem is known under the name of “separability or entanglement problem”, and has been a subject of intensive studies in the recent years.Simple entanglement criteriaThe difficulty of the separability problem comes from the fact that rho admits in general an infinite number of decompositions into a mixture of some states, and one has to check whether among them there exists at least one of the form (*). One of the most powerful necessary conditions for separability has been found by one of the fathers of quantum information, the late Asher Peres. Peres (Technion, Haifa) observed that since Alice and Bob may prepare separable states using local operations, Alice may safely reverse the time arrow in her system, which will change the state, but will not produce something unphysical. In general, such a partial time reversal is not a physical operation, and can transform a density operator (which is positive definite) into an operator that is no more positive definite. In fact this is what happens with all pure entangled states. Mathematically speaking partial time reversal corresponds to partial transposition of the density matrix (only on Alice's side). We arrive in this way at the Peres criterion: If a stateρis separable then its partial transposition has to be positive definite.This criterion is usually called positive partial transpose condition, or shortly PPT condition. Amazingly, the PPT condition is not only necessary for separability, but it is also a sufficient condition for low dimensional systems such as two qubits (dimension 2x2)and a system composed of one qubit and one qutrit (dimension 2x3). In higher dimensions, starting from 2x4 and 3x3, this is no longer true: there exist entangled states with positive partial transpose, which are called PPT entangled states.There exist several other necessary or sufficient separability criteria which have been established and frequently discussed in recent years. For example, states that are close to the completely chaotic state (whose density operator is equal to the normalized identity) are necessarily separable. There exist also other criteria that employ entropic inequalities, uncertainty relations, or an appropriate reordering of the density matrix (so called realignment criterion) etc. There exists, however, no general simple operational criterion of separability that would work in systems of arbitrary dimension.Entanglement witnessesThe set of all states P is obviously compact and convex. If ρ1 and ρ2are legitimate states,so is their convex mixture. The set of separable states S is also compact and convex (seeFigure 1). From the theory of convex sets and Hahn-Banach theorem we conclude that for any entangled state there exists a hyperplane in the space of operators separating rhofrom S. Such a hyperplane defines uniquely a Hermitian operator W (observable) which has the following properties: The expectation value of W on all separable states, <W> ≥ 0, whereas its expectation value on ρ is negative, i.e. <W>ρ< 0.Figure 1Such an observable is for obvious reasons called entanglement witness, since it “detects” the entanglement of ρ. Every entangled state has its witnesses; the problem obviously is to find appropriate witnesses for a given state. To find out whether a given state is separable one should check whether its expectation value is non-negative for all witnesses. Obviously this is a necessary and sufficient separability criterion, but unfortunately it is not operational, in the sense that there is no simple procedure to test for all witnesses.Nevertheless, witnesses provide a very useful tool to study entanglement, especially if one has some knowledge about the state in question. They provide a sufficient entanglement condition, and may be obviously optimized (see Figure 2) by shifting the hyperplane in a parallel way towards S.Figure 2Bell inequalitiesAfter introducing the concept of separability and entanglement for mixed states, it is legitimate to ask what is the relation of mixed state entanglement and the existence of a LHV model, which requires that the state cannot violate any of the Bell-like inequalities. Let us discuss an example of such inequalities, the so called Clauser-Horne-Shimony- Holt inequality for two qubits. Let us assume that Alice and Bob measure two binary observables each, namely A 1, A 2, and B 1, B 2. The observables are random variables taking the values +1 or − 1, correlated possibly through some dependence on local hidden variables. It is easy to see that in the classical world, if B 1 + B 2 is zero, then B 1 − B 2 is either +2 or −2, and vice versa. Therefore if we define s = A 1(B 1 + B 2 ) + A 2 (B 1 − B 2 ) , we obtain that 2 ≥ s ≥ −2. This inequality holds also after averaging over various realizations. On the other hand, it can be shown that by taking suitable sets of observables for Alice and Bob we can find pure and even mixed quantum states that violate this inequality.Are Bell-like inequalities similar in this respect to witnesses, i.e. for a given entangled state can one always find a Bell-like inequality that “detects” it? The answer to this question is no, and has been already given by R. Werner in 1989. Even for two qubits there exist entangled states that admit an LHV model, i.e. cannot violate any Bell-like inequality.This observation indicates already that there is more structure in the “eggs” of Figure 1 and Figure 2. Separable states are evidently inside the PPT egg, according to the Peres condition. They admit an LHV model, i.e. they are also inside the LHV egg. But what about PPT entangled states? Do they violate some Bell-like inequality? Peres has formulated a conjecture that this not the case, and there is a lot of evidence that this conjecture is correct, although a rigorous proof is still missing.The distillability problem and bound entanglement Above we have classified quantum states according to the property of being either separable or entangled. An alternative classification approach is based on the possibility of distilling the entanglement of a given state. In a distillation protocol the entanglement of a given state is increased by performing local operations and classical communication on a set of identically prepared copies. In this way one obtains fewer, but “more entangled”, copies. This kind of technique was originally proposed in 1996 by Bennett and coworkers in the context of quantum teleportation, in order to achieve faithful transmission of quantum states over noisy channels. It also has applications in quantum cryptography as a method for quantum privacy amplification in entanglement based protocols in the presence of noise, as pointed out by David Deutsch and coworkers from Oxford.The distillability problem poses the question whether a given quantum state can be distilled or not. A separable state can never be distilled because the average entanglement of a set of states cannot be increased by local operations. Furthermore, the positivity of the partial transpose ensures that no distillation is possible. Thus, a given PPT entangled state is not distillable, and is therefore called bound entangled. There mayeven exist undistillable entangled states which do not have the PPT property. However, this conjecture is not proved at the moment.The first example of a PPT entangled state has been found by Pawel Horodecki from Gdansk in 1997. These states are so called edge states, which means that they cannot be written as a mixture of a separable state and a PPT entangled state. Particularly simple families of states have been suggested by Charles Bennett and coworkers at IBM, New York. They have found the so called unextendible product bases (UPB), i.e. sets of orthogonal product state-vectors, with the property that the space orthogonal to this set does not contain any product vector. It turns out that the projector onto this space is a PPT state, which obviously has to be entangled since it does not contain any product vector in its range (note that all state-vectors in the decomposition of a separable state ρinto a mixture of product states belong automatically to the range of ρ).The existence of bound entanglement is a mysterious invention of Nature. It is an interesting question to ask whether bound entanglement is a useful resource to perform quantum information processing tasks. It was shown so far that this is not the case for communication protocols such as quantum teleportation and quantum dense coding (i.e.a protocol that allows to enhance the transmission of classical information, using entanglement). However, surprisingly, it is possible to distill a secret key in quantum cryptography, starting from certain bound entangled states.Entanglement detectionAs discussed above, entanglement is a precious resource in quantum information processing. Typically in a real world experiment noise is always present and it leads to a decrease of entanglement in general. Thus, it is of fundamental interest for experimental applications to be able to test the entanglement properties of the generated states. A traditional method to this aim is represented by the Bell inequalities, a violation of which indicates the presence of entanglement. However, as mentioned above, not every entangled state violates a Bell inequality. So, not all entangled states can be detected by using this method.Another possibility is to perform complete state tomography, which allows determining all the elements of the density matrix. This is a useful method to get a complete knowledge of the density operator of a quantum system, but to detect entanglement it is an expensive process as it requires an unnecessary large number of measurements. If one has certain knowledge about the state the most appropriate technique is the measurement of the witness observable, which can be achieved by few local measurements. A negative expectation value clearly indicates the presence of entanglement.All these methods have been successfully implemented in various experiments. Recently another method for the detection of entanglement was suggested based on the physical approximation of the partial transpose. It remains a challenge to implement this idea in the laboratory because it requires the implementation of non local measurements.Entanglement measuresWhen classifying a quantum state as being entangled, a natural question is to quantify the amount of entanglement it contains. For pure quantum states there exists a well defined entanglement measure, namely the von Neumann entropy of the density operator of a subsystem of the composite state. For mixed states the situation is more complicated. There are several different possibilities to define an entanglement measure. The so called entanglement cost describes the amount of entanglement one needs in order to generate a given state. An alternative measure is the entanglement of formation, which is a more abstract definition. A further possibility to quantify entanglement is given by the minimum distance to separable states. Finally, motivated by physical applications, one can introduce the distillable entanglement which quantifies the extractable amount of entanglement.Unfortunately all of these quantities are very difficult to compute in general. For example, in order to determine the entanglement of formation one has to find the decomposition of the state that leads to the minimum average von Neumann entropy of a subsystem and this is a very challenging task. So far a complete analytical formula for the entanglement of formation only exists for composite systems of two qubits.Entanglement in multipartite systemsSo far, we have restricted ourselves to the case of composite systems with two subsystems, so called bipartite systems. When considering more than two parties, i.e multipartite systems, the situation becomes much more complex. For example, for the most simple tripartite case of three qubits, a pure state can be either completely separable, or biseparable (i.e. one of the three parties is not entangled with the other two), or genuinely entangled among all three parties. The latter class again consists of inequivalent subclasses, the so called GHZ and W states. This concept can be generalized to mixed states. For more than three parties it is easy to imagine that the number of subclasses grows fast.In recent years there has been much progress in the creation of multipartite entangled states in the laboratory. The existence of genuine multipartite entanglement has also been demonstrated experimentally by using the concept of witness operators.Even if the full classification of multipartite entanglement is a formidable task, certain classes of states, the so called graph states, have been completely characterized and shown to be useful both for quantum computational and quantum error correction protocols. Moreover, a deeper understanding of entanglement has proved to be very fruitful in connection with statistical properties of physical systems. All of these problems are discussed in more details in other sections of this publication.References[1] Einstein, P. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935)[2] J.S. Bell, Physics 1, 195 (1964)[3] P. Horodecki, Phys. Lett. A 232, 333 (1997)[4] M. Lewenstein et al., J. Mod. Opt. 47, 2481 (2000)[5] A. Peres, Phys. Rev. Lett. 77, 1413 (1996)[6] E. Schrödinger, Naturwissenschaften 23, 807 (1935)[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989) Contact information of the author of this article Maciej LewensteinInstitut de Ciènces Fotòniques (ICFO)C/Jordi Girona 29, Nexus 2908034 BarcelonaSpainEmail: maciej.lewenstein@icfo.esChiara MacchiavelloIstituto Nazionale di Fisicadella Materia, Unita' di Pavia Dipartimento di Fisica "A. Volta"via Bassi 6I-27100 PaviaItalyEmail: chiara@unipv.itProf. Dr. Dagmar BrussInst. fuer Theoretische Physik IIIHeinrich-Heine-Universitaet Duesseldorf Universitaetsstr. 1, Geb. 25.32D-40225 Duesseldorf,GermanyEmail: bruss@thphy.uni-duesseldorf.de。

利用冯· 诺依曼熵获得最大纠缠态的形式

利用冯诺依曼熵获得最大纠缠态的形式朱孟正;赵春然;李洪俊;张东杰【摘要】纠缠在量子信息处理中有许多重要的应用,正如Bell态对量子通信的实施是必不可少的.考虑如何得到Bell态,本文提出了一种用冯· 诺依曼熵求解二体或三体系统中最大纠缠态表示形式的方法.计算二体或三体系统的量子态的冯· 诺依曼熵,并将约化密度算符与用Bloch矢量表示的密度算符进行比较.根据密度算符具有正的、厄密性的特点,得到了最大纠缠态解析式,如Bell态和GHZ态.【期刊名称】《吉林师范大学学报（自然科学版）》【年(卷),期】2018(039)002【总页数】5页(P78-82)【关键词】冯·诺依曼熵;纠缠态;密度算符【作者】朱孟正;赵春然;李洪俊;张东杰【作者单位】淮北师范大学物理与电子信息学院,安徽淮北235000;淮北师范大学信息学院,安徽淮北235000;淮北师范大学物理与电子信息学院,安徽淮北235000;淮北师范大学物理与电子信息学院,安徽淮北235000;淮北师范大学物理与电子信息学院,安徽淮北235000【正文语种】中文【中图分类】O413.10 IntroductionThe key feature of quantum mechanics that lies behind quantum information theory is quantum entanglement.Quantum entanglement refers to correlations between the results of measurements made on distinct subsystems of a composite system that can not be explained in terms of standard statistical correlations between classical properties inherent in each subsystem.For the bipartite quantum systems，a correlation between two subsystems is simply the statement that if a measurement of one subsystem yields the result A then a measurement on the second subsystem will yield the result B with some probability.Perfect correlation occurs when the second result is certain，given the outcome of the first[1-2].It has become clear that entanglement is a new quantum resource for tasks that cannot be performed by means of classical resources.It can be manipulated，broadcast，controlled and distributed.Remarkably，entanglement is a resource which，though it does not carry information itself，can help in such tasks as the reduction of classical communication complexity，entanglement-assisted orientation in space，quantum estimation of a damping constant，frequency standards improvement，and clock synchronization.Entanglement plays a fundamental role in quantum communication between parties separated by macroscopic distances[3-5].For these tasks，the maximally entangled state is an indispensable quantum resource.The Bell states are the “canonical” maximally entangled states in the bipartite systems[6].Have we ever considered how the Bell states are given? This article presents a method for solving the representation of the maximally entangled states intwo-particle systems using von Neumann entropy.Then we generalize this approach to the three-body entanglement problem[7].1 Von Neumann entropy and the maximally entangled stateFor any pure state of two parties，for instance，a pure state of two qubits can be written as|ψ〉AB=c0|00〉+c1|01〉+c2|10〉+c3|11〉.(1)A unique measure of bipartite entanglement for pure states is given by the partial von Neumann entropy.The von Neumann entropy of a state is defined asS(ρ)=-Tr(ρlogρ)，(2)where the symbol ρ is the density operator for the system and Tr(…) denotes the trace operation.We can obtain the density operator associated with the quantum state in Eq.(1)，(3)The entanglement of the partly entangled pure state in Eq.(3) can be naturally parametrized by its entropy of entanglement[8]，defined as the von Neumann entropy of either ρA or ρB，S(ρAB)=S(ρA)=S(ρB).(4)We choose the standard basis to calculate the partial trace.For the densityoperator in Eq.(3)，we can obtain(5)where TrB refers to the partial trace over mode B.Analogously，(6)We diagonalise ρA or ρB.When the reduced density operator ρ is written in this diagonal form，our von Neumann entropy in Eq.(2) becomes(7)where the symbols ρn are the associated(non-negative) eigenvalues of the reduced density operator ρ diagonalised，which sum to unity，that is to the diagonalization，the reduced density operator ρA and ρB can be written as(8)where an d ζ≡4|c1c2-c0c3|2.Obviously，0≤ζ≤1.By making use of the diagonal representation of the reduced density operator in Eq.(8)，we write von Neumann entropy in the form：S(ρAB)=S(ρA)=S(ρB)=-ρ+logρ+-ρ-logρ-.(9)Any two-by-two matrix can be written as a weighted sum of the four Pauli operators.This means，in turn，that any operator associated with our qubit can also be expressed in terms of these operators.In particular，we can write the density operator in the form：(10)where I is the identity operator，r=(u，ν，w) is a Bloch vector，and σ=(σx，σy，σz) is the vector operator[9].Here Eq.(10)，the factor 1/2 ensures that Tr(ρ)=1.The density operator ρ is a positive Hermitian operator.The Hermiticity of ρ ensures that u，ν，and w are real.For the two-state system，the density operator of Eq.(10) can be written in the diagonal form：ρ=ρ+|ρ+〉〈ρ+|+ρ-|ρ-〉〈ρ-| .(11)where the states |ρ+〉and |ρ-〉are the eigenvectors of ρ correspondingto the eigenvalues Neumann entropy can be written as the equation(9)，but ρ± are described by the variable r.We have delineated Figure 1 about the von Neumann entropy as a function of the variable r.The positivity of the density operator ρ requires that u2+ν2+w2≤1.It is worth noting that the eigenvectors of ρ，namely |ρ+〉and |ρ-〉，are also the eigenvec tors of the operator r·σ corresponding to the eigenvalues ±r.If the vector’s tip of r lies on the surface of the Bloch sphere(r=1)，the diagonal density operator in Eq.(11) reduces the pure state |ρ+〉〈ρ+|.Ifr<1，the Bloch vector describes a point inside the Bloch sphere and corresponds to a mixed state.The farther the point is from the surface，the higher the degree of mixing of the mixed state is.This is to say，the resultis a more mixed state with a greater entropy.When the entropy of the subsystem reduced states are maximal，such states are called maximally entangled.The maximally entangled state of two subsystems associatedwith Eq.(1) requires r=0 from Fig.1.Fig.1 The von Neumann entropy versus the variable rAccording to the condition r=0，we can obtain the real parameteru=ν=w=0 due to the Hermiticity of ρ.Compared Eq.(5) and Eq.(6)with Eq.(10)，we can also write the following relationship：|c0|2+|c1|2-|c2|2-|c3|2=0，|c0|2-|c1|2+|c2|2-|c3|2=0，|c0|2+|c1|2+|c2|2+|c3|2=1，|c1c2-c0c3|=1/2(12)in the condition of the maximally entangled state.We choose the real coefficients c0，c1，c2，and c3 for simplicity.The individual equation in Eqs.(12) is not linearly independent of the other，for example，the last equation.By solving Eqs.(12)，we can obtain the coefficients of Eq.(1) for the maximally entangled state as follows.If and c1=c2=0；if andc0=c3=0.This result exactly corresponds to the Bell states for the maximally entangled subsystems.The four Bell states are conventionally written in the form(13)They are known as the four maximally entangled two-qubit Bell states.The von Neumann entropy of this density operator of the Bell states is positive and maximal.For the quantum state of two qubits，the Bell states of have a special prominence.The reasons for this include their simplicity and the fact that they have been realized in a number of diverse experiments.Just as Bell states are essential for the implementation of quantumcommunication with perfect fidelity，the importance of such a state in the distribution of bipartite entanglement is obvious.Certainly，multipartite maximally entangled states also play many crucial roles in quantum computation and quantum communication[10].Then we this approach is also generalized to solve the maximally entangled states of the three-body entanglement.For any pure state of tripartite，for instance，a pure state of three qubits can be written as|ψ〉ABC=c0|000〉+c1|001〉+c2|010〉+c3|011〉+c4|100〉+c5|101〉+c6|110〉+c7|111〉.(14)We can obtain the density operator associated with the quantum tripartite state in Eq.(14)，ρABC=|ψ〉ABC〈ψ|.(15)For the tripartite density operator，we choose the standard basis to calculate the partial trace in order to obtain(16)whereρA11≡|c0|2+|c1|2+|c2|2+|c3|2，ρB11≡|c0|2+|c1|2+|c4|2+|c5|2，ρC11≡|c0|2+|c2|2+|c4|2+|c6|2，Compared Eqs.(16) with Eq.(10) in the condition of r=0，we can obtain the coefficients of Eq.(14) in order to write the maximally entangled state of tripartite system as follows(17)Analogously，for simplicity we have chosen the real coefficients ci(i=0，1，2，…，7).They are known as the eight maximally entangled Greenberger-Horne-Zeilinger(GHZ)states[11].The GHZ state is a certain type of entangled state that involves at least three subsystems.The GHZ states are used in several protocols in quantum communication and cryptography，for example，in secret sharing.Thus the GHZ-state and the W-state represent two very different kinds of tripartite entanglement.In a certain sense，the W-state is “less entangled” than the GHZ-state.Therefore，the W-state does not belong to our solution.2 ConclusionThe degree which state in a quantum system consisting of two “particles” is entangled is measured by the von Neumann entropy of either of the two reduced density operators of the state.When the entropy of the subsystem reduced states are maximal，such states are called maximally entangled.We calculate the von Neumann entropy of the pipartite or tripartite systems and compare the reduced density operator with the density operator in the form of the Bloch vertor.According to the character that the density operator ρ is a positive Hermitian operator，we obtain the maximally entangled states such as the Bell states and the GHZ states. References【相关文献】[1]GERRY C C，KNIGHT P L.Introductory quantum optics[M].Cambridge：Cambridge University Press，2005.[2]PAN J W，CHEN Z B，LU C Y，et al.Multiphoton entanglement andinterferometry[J].Rev Mod Phys，2012，84(2)：777-838.[3]HORODECKI R，HORODECKI P L，HORODECKI M L，et al.Quantumentanglement[J].Rev Mod Phys，2009，81(2)：865-942.[4]SCHMID C，KIESEL N，WEBER U K，et al.Quantum teleportation and entanglement swapping with linear optics logic gates[J].New J Phys，2009，11：33008-1-33008-10. [5]ZHANG X L，WANG M L，YANG L L.Hawk-dovegame model of quantum under asymmetric information[J].Journal of Jilin Normal University(Natural Science Edition)，2011，32(4)：8-12.[6]EISERT J，CRAMER M，PLENIO M B.Colloquium：Area laws for the entanglement entropy[J].Rev Mod Phys，2010，82(1)：277-306.[7]AMICO L，FAZIO R，OSTERLOH A，et al.Entanglement in many-body systems[J].Rev Mod Phys，2008，80(2)：517-576.[8]BENNETT C H，BERNSTEIN H J，POPESCU S，et al.Concentrating partial entanglement by local operations[J].Phys Rev A，1996，53：2046-2052.[9]BARNETT S M.Quantum information[M].New York：Oxford University Press，2009.[10]DÜR W.Multipartit e entanglement that is robust against disposal of particles[J].Phys Rev A，2001，63(2)：020303-1-020303-4.[11]GREENBERGER D M，HORNE M A，SHIMONY A，et al.Bell’s theorem without inequalities[J].Am J Phys，1990，58(12)：1131-1143.。

量子信息和量子纠缠理论

Multipartite Schmidt-correlated State
Fully separable
PPT
Fully separable (maximally entangled)
~ 1 (N)
M.J. Zhao, S.M. Fei and Z.X. Wang, Phys. Lett. A 372(2008)2552
S. Albeverio, S.M. Fei, Phys. Lett. A 276(2000)8 S. Albeverio, S.M. Fei and W.L. Yang, Comm. Theor. Phys. 38 (2002) 301
S. Albeverio, S.M. Fei and W.L. Yang, Phys. Rev. A 66 (2002) 012301 M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A 60, 1888 (1999)
Separable! Separable！
Separability of mixed states: no general criteria a) Peres (PPT) criterion:
Peres PRL 77, 1413 (1996)
2x2, 2x3:
PPT
Separable
Horodeckis, Phys. Lett. A 223,1 (1996)
Caltech (Kimble et al)
/~qoptics/teleport.html
Nature 390(1997) 575
Science 282(1998) 706
Wigner functions before &after

量子相变中的量子纠缠度量研究-英文版

RG flow
• c-theorem:
cUV cIR
fixed point
(Zamolodchikov, 1986)
UV
fixed point
IR
• Massive theory (off critical) Block entropy saturation
fixed point
RG flow
UV
Irreversibility of RG trajectories
Loss of entanglement
Open Systems & Quantum Information Milano, 10 Marzo 2006
• Local Entropy: when the subsystem A is a single site.
• The concurrence measures the entanglement between two sites after having traced out the remaining sites. • The transition is signaled by the first derivative of the concurrence, which diverges logarithmically (specific heat).

h 1 lnh 1
S1 has the same singularity as
Local (single site) entropy: 1 ρ1 = I + σ z σ z S1 = Tr ρ1lnρ1 2
σz

Local measures of entanglement based on the 2-site density matrix depend on 2-point functions

代数英语

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

两体非马尔科夫环境下的quantum discord研究

两体非马尔科夫环境下的quantum discord研究摘要研究了两个相互独立的原子在非马尔科夫环境下的动力学。

发现在一些初态下随着参数（原子频率与边带频率的失谐量）减小到一定程度后quantum discord会衰减到一个稳定值，并且这个稳定值会随着参数的减小而增大。

这说明在一段时间内耗散能够被有效的抑制。

在部分初态下，concurrence消失而quantum discord依然保持在一定数值，说明此时quantum discord所描述的是非纠缠的部分，此外quantum discord中的非纠缠部分时间演化受到所选取的初态的影响。

关键字量子关联；量子纠缠；quantum discord中图分类号o431 文献标识码a 文章编号 1674-6708（2013）94-0111-02随着研究的深入发现量子纠缠在描述相关性等方面还有着许多的不足之处，为了能够更好的衡量量子关联，2001年提出了一个新的概念quantum discord [1]。

一开始quantum discord的出现并没有得到足够的重视，大量的研究还是主要围绕着量子纠缠来进行的，直到近近几年一些研究的突破进展，quantum discord才逐渐受到重视，相关的研究也展开了。

在本文中我们使用discord 来研究两独立原子在非马尔科夫环境下的力学。

当我们选取初态（）和态时发现quantum discord在此种非马尔科夫环境下，也能像concurrence 一样衰减到一个稳定值，并且两者的演化趋势相似。

当初态为（）时concurrence消失，quantum discord依然存在并且稳定在一定的数值上。

1两体非马尔科夫环境下的quantum discord我们选取两个相互独立的二能级原子1和2，把它们分别放置在两个非马尔科夫库环境中（此处我们选择光子晶体这种非马尔科夫环境），最初为真空态。

我们可以得到单原子子系统的哈密顿量和各向异性光子晶体下的色散关系，参数是原子频率与边带频率的失谐量，（1）是一个跟频带属性相关的常数，其中是原子偶极距，是真空介电常数[2] [3] [4]。

不对称外磁场下两量子比特系统的几何相

不对称外磁场下两量子比特系统的几何相苏耀恒;陈爱民;王军;李跃文【摘要】研究了不对称旋转外磁场下具有 XXZ型海森堡相互作用的两量子比特系统的几何相。

考虑体系的绝热条件，利用数值模拟的方法得到量子比特系统的4个本征态的Berry相，研究了外加旋转磁场的极角以及量子比特之间相互作用的各向异性参数对4个本征态的Berry相的影响。

研究结果表明：当极角保持不变，各向异性参数由0增加至无穷大的过程中，系统的哈密顿量由一种极限下的含外场的 XX 模型经过中间的海森堡模型，逐渐演化为另外一种极限下的 Ising模型。

4个本征态的 Berry 相都有各自独特的变化规律，且极角越小几何相趋于稳定越快。

通过对系统 Berry相的研究，可以得到系统在不同参数区间对应的模型的转化，并对本征态的几何性质有更进一步的认识。

【期刊名称】《河南科技大学学报（自然科学版）》【年(卷),期】2017(038)002【总页数】5页(P79-83)【关键词】量子信息;两量子比特系统;几何相;海森堡相互作用【作者】苏耀恒;陈爱民;王军;李跃文【作者单位】西安工程大学理学院，陕西西安 710048;西安工程大学理学院，陕西西安 710048; 西安交通大学理学院，陕西西安 710049;西安工程大学理学院，陕西西安 710048;中航光电科技股份有限公司光电设备事业部，河南洛阳471000【正文语种】中文【中图分类】O469量子信息[1]是指在物理系统的量子态中所保存的物理信息。

量子信息最基本的单元是量子比特[2]，这是一个二能级态的量子系统。

例如，光子的两个偏振方向、原子中电子的两个能级或者环路中电流的不同方向等，在测量时都可以很容易被区分开来。

量子系统的哈密顿量不仅决定了量子态的能级，更决定了这个物理系统的态随时间的演化情况。

在许多应用中，哈密顿量的物理参数都是由含时的外部或环境因素决定的，因而研究含时的哈密顿量在实际的物理领域中是很重要的。

近藤效应

Tunable Kondo eﬀect in a single donor atomnsbergen 1,G.C.Tettamanzi 1,J.Verduijn 1,N.Collaert 2,S.Biesemans 2,M.Blaauboer 1,and S.Rogge 11Kavli Institute of Nanoscience,Delft University of Technology,Lorentzweg 1,2628CJ Delft,The Netherlands and2InterUniversity Microelectronics Center (IMEC),Kapeldreef 75,3001Leuven,Belgium(Dated:September 30,2009)The Kondo eﬀect has been observed in a single gate-tunable atom.The measurement device consists of a single As dopant incorporated in a Silicon nanostructure.The atomic orbitals of the dopant are tunable by the gate electric ﬁeld.When they are tuned such that the ground state of the atomic system becomes a (nearly)degenerate superposition of two of the Silicon valleys,an exotic and hitherto unobserved valley Kondo eﬀect appears.Together with the “regular”spin Kondo,the tunable valley Kondo eﬀect allows for reversible electrical control over the symmetry of the Kondo ground state from an SU(2)-to an SU(4)-conﬁguration.The addition of magnetic impurities to a metal leads to an anomalous increase of their resistance at low tem-perature.Although discovered in the 1930’s,it took until the 1960’s before this observation was satisfactorily ex-plained in the context of exchange interaction between the localized spin of the magnetic impurity and the de-localized conduction electrons in the metal [1].This so-called Kondo eﬀect is now one of the most widely stud-ied phenomena in condensed-matter physics [2]and plays a mayor role in the ﬁeld of nanotechnology.Kondo ef-fects on single atoms have ﬁrst been observed by STM-spectroscopy and were later discovered in a variety of mesoscopic devices ranging from quantum dots and car-bon nanotubes to single molecules [3].Kondo eﬀects,however,do not only arise from local-ized spins:in principle,the role of the electron spin can be replaced by another degree of freedom,for example or-bital momentum [4].The simultaneous presence of both a spin-and an orbital degeneracy gives rise to an exotic SU(4)-Kondo eﬀect,where ”SU(4)”refers to the sym-metry of the corresponding Kondo ground state [5,6].SU(4)Kondo eﬀects have received quite a lot of theoret-ical attention [6,7],but so far little experimental work exists [8].The atomic orbitals of a gated donor in Si consist of linear combinations of the sixfold degenerate valleys of the Si conduction band.The orbital-(or more speciﬁ-cally valley)-degeneracy of the atomic ground state is tunable by the gate electric ﬁeld.The valley splitting ranges from ∼1meV at high ﬁelds (where the electron is pulled towards the gate interface)to being equal to the donors valley-orbit splitting (∼10-20meV)at low ﬁelds [9,10].This tunability essentially originates from a gate-induced quantum conﬁnement transition [10],namely from Coulombic conﬁnement at the donor site to 2D-conﬁnement at the gate interface.In this article we study Kondo eﬀects on a novel exper-imental system,a single donor atom in a Silicon nano-MOSFET.The charge state of this single dopant can be tuned by the gate electrode such that a single electron (spin)is localized on the pared to quantum dots (or artiﬁcial atoms)in Silicon [11,12,13],gated dopants have a large charging energy compared to the level spac-ing due to their typically much smaller size.As a result,the orbital degree of freedom of the atom starts to play an important role in the Kondo interaction.As we will argue in this article,at high gate ﬁeld,where a (near)de-generacy is created,the valley index forms a good quan-tum number and Valley Kondo [14]eﬀects,which have not been observed before,appear.Moreover,the Valley Kondo resonance in a gated donor can be switched on and oﬀby the gate electrode,which provides for an electri-cally controllable quantum phase transition [15]between the regular SU(2)spin-and the SU(4)-Kondo ground states.In our experiment we use wrap-around gate (FinFET)devices,see Fig.1(a),with a single Arsenic donor in the channel dominating the sub-threshold transport charac-teristics [16].Several recent experiments have shown that the ﬁngerprint of a single dopant can be identiﬁed in low-temperature transport through small CMOS devices [16,17,18].We perform transport spectroscopy (at 4K)on a large ensemble of FinFET devices and select the few that show this ﬁngerprint,which essentially consists of a pair of characteristic transport resonances associ-ated with the one-electron (D 0)-and two-electron (D −)-charge states of the single donor [16].From previous research we know that the valley splitting in our Fin-FET devices is typically on the order of a few meV’s.In this Report,we present several such devices that are in addition characterized by strong tunnel coupling to the source/drain contacts which allows for suﬃcient ex-change processes between the metallic contacts and the atom to observe Kondo eﬀects.Fig.1b shows a zero bias diﬀerential conductance (dI SD /dV SD )trace at 4.2K as a function of gate volt-age (V G )of one of the strongly coupled FinFETs (J17).At the V G such that a donor level in the barrier is aligned with the Fermi energy in the source-drain con-tacts (E F ),electrons can tunnel via the level from source to drain (and vice versa)and we observe an increase in the dI SD /dV SD .The conductance peaks indicated bya r X i v :0909.5602v 1 [c o n d -m a t .m e s -h a l l ] 30 S e p 2009FIG.1:Coulomb blocked transport through a single donor in FinFET devices(a)Colored Scanning Electron Micrograph of a typical FinFET device.(b)Diﬀerential conductance (dI SD/dV SD)versus gate voltage at V SD=0.(D0)and(D−) indicate respectively the transport resonances of the one-and two-electron state of a single As donor located in the Fin-FET channel.Inset:Band diagram of the FinFET along the x-axis,with the(D0)charge state on resonance.(c)and(d) Colormap of the diﬀerential conductance(dI SD/dV SD)as a function of V SD and V G of samples J17and H64.The red dots indicate the(D0)resonances and data were taken at1.6 K.All the features inside the Coulomb diamonds are due to second-order chargeﬂuctuations(see text).(D0)and(D−)are the transport resonances via the one-electron and two-electron charge states respectively.At high gate voltages(V G>450mV),the conduction band in the channel is pushed below E F and the FET channel starts to open.The D−resonance has a peculiar double peak shape which we attribute to capacitive coupling of the D−state to surrounding As atoms[19].The current between the D0and the D−charge state is suppressed by Coulomb blockade.The dI SD/dV SD around the(D0)and(D−)resonances of sample J17and sample H64are depicted in Fig.1c and Fig.1d respectively.The red dots indicate the po-sitions of the(D0)resonance and the solid black lines crossing the red dots mark the outline of its conducting region.Sample J17shows aﬁrst excited state at inside the conducting region(+/-2mV),indicated by a solid black line,associated with the valley splitting(∆=2 mV)of the ground state[10].The black dashed lines indicate V SD=0.Inside the Coulomb diamond there is one electron localized on the single As donor and all the observable transport in this regionﬁnds its origin in second-order exchange processes,i.e.transport via a vir-tual state of the As atom.Sample J17exhibits three clear resonances(indicated by the dashed and dashed-dotted black lines)starting from the(D0)conducting region and running through the Coulomb diamond at-2,0and2mV. The-2mV and2mV resonances are due to a second or-der transition where an electron from the source enters one valley state,an the donor-bound electron leaves from another valley state(see Fig.2(b)).The zero bias reso-nance,however,is typically associated with spin Kondo eﬀects,which happen within the same valley state.In sample H64,the pattern of the resonances looks much more complicated.We observe a resonance around0mV and(interrupted)resonances that shift in V SD as a func-tion of V G,indicating a gradual change of the internal level spectrum as a function of V G.We see a large in-crease in conductance where one of the resonances crosses V SD=0(at V G∼445mV,indicated by the red dashed elipsoid).Here the ground state has a full valley degen-eracy,as we will show in theﬁnal paragraph.There is a similar feature in sample J17at V G∼414mV in Fig.1c (see also the red cross in Fig.1b),although that is prob-ably related to a nearby defect.Because of the relative simplicity of its diﬀerential conductance pattern,we will mainly use data obtained from sample J17.In order to investigate the behavior at the degeneracy point of two valley states we use sample H64.In the following paragraphs we investigate the second-order transport in more detail,in particular its temper-ature dependence,ﬁne-structure,magneticﬁeld depen-dence and dependence on∆.We start by analyzing the temperature(T)dependence of sample J17.Fig.2a shows dI SD/dV SD as a function of V SD inside the Coulomb diamond(at V G=395mV) for a range of temperatures.As can be readily observed from Fig.2a,both the zero bias resonance and the two resonances at V SD=+/-∆mV are suppressed with increasing T.The inset of Fig.2a shows the maxima (dI/dV)MAX of the-2mV and0mV resonances as a function of T.We observe a logarithmic dependence on T(a hallmark sign of Kondo correlations)at both resonances,as indicated by the red line.To investigate this point further we analyze another sample(H67)which has sharper resonances and of which more temperature-dependent data were obtained,see Fig.2c.This sample also exhibits the three resonances,now at∼-1,0and +1mV,and the same strong suppression by tempera-ture.A linear background was removed for clarity.We extracted the(dI/dV)MAX of all three resonances forFIG.2:Electrical transport through a single donor atom in the Coulomb blocked region(a)Diﬀerential conductance of sample J17as a function of V SD in the Kondo regime(at V G=395mV).For clarity,the temperature traces have been oﬀset by50nS with respect to each other.Both the resonances with-and without valley-stateﬂip scale similarly with increasing temperature. Inset:Conductance maxima of the resonances at V SD=-2mV and0mV as a function of temperature.(b)Schematic depiction of three(out of several)second-order processes underlying the zero bias and±∆resonances.(c)Diﬀerential conductance of sample H67as a function of V SD in the Kondo regime between0.3K and6K.A linear(and temperature independent) background on the order of1µS was removed and the traces have been oﬀset by90nS with respect to each other for clarity.(d)The conductance maxima of the three resonances of(c)normalized to their0.3K value.The red line is aﬁt of the data by Eq.1.all temperatures and normalized them to their respective(dI/dV)MAX at300mK.The result is plotted in Fig.2d.We again observe that all three peaks have the same(log-arithmic)dependence on temperature.This dependenceis described well by the following phenomenological rela-tionship[20](dI SD/dV SD)max (T)=(dI SD/dV SD)T 2KT2+TKs+g0(1)where TK =T K/√21/s−1,(dI SD/dV SD)is the zero-temperature conductance,s is a constant equal to0.22 [21]and g0is a constant.Here T K is the Kondo tem-perature.The red curve in Fig.2d is aﬁt of Eq.(1)to the data.We readily observe that the dataﬁt well and extract a T K of2.7K.The temperature scaling demon-strates that both the no valley-stateﬂip resonance at zero bias voltage and the valley-stateﬂip-resonance atﬁnite bias are due to Kondo-type processes.Although a few examples ofﬁnite-bias Kondo have been reported[15,22,23],the corresponding resonances (such as our±∆resonances)are typically associated with in-elastic cotunneling.Aﬁnite bias between the leads breaks the coherence due to dissipative transitions in which electrons are transmitted from the high-potential-lead to the low-potential lead[24].These dissipative4transitions limit the lifetime of the Kondo-type processes and,if strong enough,would only allow for in-elastic events.In the supporting online text we estimate the Kondo lifetime in our system and show it is large enough to sustain theﬁnite-bias Kondo eﬀects.The Kondo nature of the+/-∆mV resonances points strongly towards a Valley Kondo eﬀect[14],where co-herent(second-order)exchange between the delocalized electrons in the contacts and the localized electron on the dopant forms a many-body singlet state that screens the valley index.Together with the more familiar spin Kondo eﬀect,where a many-body state screens the spin index, this leads to an SU(4)-Kondo eﬀect,where the spin and charge degree of freedom are fully entangled[8].The ob-served scaling of the+/-∆-and zero bias-resonances in our samples by a single T K is an indication that such a fourfold degenerate SU(4)-Kondo ground state has been formed.To investigate the Kondo nature of the transport fur-ther,we analyze the substructure of the resonances of sample J17,see Fig.2a.The central resonance and the V SD=-2mV each consist of three separate peaks.A sim-ilar substructure can be observed in sample H67,albeit less clear(see Fig.2c).The substructure can be explained in the context of SU(4)-Kondo in combination with a small diﬀerence between the coupling of the ground state (ΓGS)-and theﬁrst excited state(ΓE1)-to the leads.It has been theoretically predicted that even a small asym-metry(ϕ≡ΓE1/ΓGS∼=1)splits the Valley Kondo den-sity of states into an SU(2)-and an SU(4)-part[25].Thiswill cause both the valley-stateﬂip-and the no valley-stateﬂip resonances to split in three,where the middle peak is the SU(2)-part and the side-peaks are the SU(4)-parts.A more detailed description of the substructure can be found in the supporting online text.The split-ting between middle and side-peaks should be roughly on the order of T K[25].The measured splitting between the SU(2)-and SU(4)-parts equals about0.5meV for sample J17and0.25meV for sample H67,which thus corresponds to T K∼=6K and T K∼=3K respectively,for the latter in line with the Kondo temperature obtained from the temperature dependence.We further note that dI SD/dV SD is smaller than what we would expect for the Kondo conductance at T<T K.However,the only other study of the Kondo eﬀect in Silicon where T K could be determined showed a similar magnitude of the Kondo signal[12].The presence of this substructure in both the valley-stateﬂip-,and the no valley-stateﬂip-Kondo resonance thus also points at a Valley Kondo eﬀect.As a third step,we turn our attention to the magnetic ﬁeld(B)dependence of the resonances.Fig.3shows a colormap plot of dI SD/dV SD for samples J17and H64 both as a function of V SD and B at300mK.The traces were again taken within the Coulomb diamond.Atﬁnite magneticﬁeld,the central Kondo resonances of both de-vices split in two with a splitting of2.2-2.4mV at B=FIG.3:Colormap plot of the conductance as a function of V SD and B of sample J17at V G=395mV(a)and H64at V G=464mV(b).The central Kondo resonances split in two lines which are separated by2g∗µB B.The resonances with a valley-stateﬂip do not seem to split in magneticﬁeld,a feature we associate with the diﬀerent decay-time of parallel and anti-parallel spin-conﬁgurations of the doubly-occupied virtual state(see text).10T.From theoretical considerations we expect the cen-tral Valley Kondo resonance to split in two by∆B= 2g∗µB B if there is no mixing of valley index(this typical 2g∗µB B-splitting of the resonances is one of the hall-marks of the Kondo eﬀect[24]),and to split in three (each separated by g∗µB B)if there is a certain degree of valley index mixing[14].Here,g∗is the g-factor(1.998 for As in Si)andµB is the Bohr magneton.In the case of full mixing of valley index,the valley Kondo eﬀect is expected to vanish and only spin Kondo will remain [25].By comparing our measured magneticﬁeld splitting (∆B)with2g∗µB B,weﬁnd a g-factor between2.1and 2.4for all three devices.This is comparable to the result of Klein et al.who found a g-factor for electrons in SiGe quantum dots in the Kondo regime of around2.2-2.3[13]. The magneticﬁeld dependence of the central resonance5indicates that there is no signiﬁcant mixing of valley in-dex.This is an important observation as the occurrence of Valley Kondo in Si depends on the absence of mix-ing(and thus the valley index being a good quantum number in the process).The conservation of valley in-dex can be attributed to the symmetry of our system. The large2D-conﬁnement provided by the electricﬁeld gives strong reason to believe that the ground-andﬁrst excited-states,E GS and E1,consist of(linear combi-nations of)the k=(0,0,±kz)valleys(with z in the electricﬁeld direction)[10,26].As momentum perpen-dicular to the tunneling direction(k x,see Fig.1)is con-served,also valley index is conserved in tunneling[27]. The k=(0,0,±k z)-nature of E GS and E1should be as-sociated with the absence of signiﬁcant exchange interac-tion between the two states which puts them in the non-interacting limit,and thus not in the correlated Heitler-London limit where singlets and triplets are formed.We further observe that the Valley Kondo resonances with a valley-stateﬂip do not split in magneticﬁeld,see Fig.3.This behavior is seen in both samples,as indicated by the black straight solid lines,and is most easily ob-served in sample J17.These valley-stateﬂip resonances are associated with diﬀerent processes based on their evo-lution with magneticﬁeld.The processes which involve both a valleyﬂip and a spinﬂip are expected to shift to energies±∆±g∗µB B,while those without a spin-ﬂip stay at energies±∆[14,25].We only seem to observe the resonances at±∆,i.e.the valley-stateﬂip resonances without spinﬂip.In Ref[8],the processes with both an orbital and a spinﬂip also could not be observed.The authors attribute this to the broadening of the orbital-ﬂip resonances.Here,we attribute the absence of the processes with spinﬂip to the diﬀerence in life-time be-tween the virtual valley state where two spins in seperate valleys are parallel(τ↑↑)and the virtual state where two spins in seperate valleys are anti-parallel(τ↑↓).In con-trast to the latter,in the parallel spin conﬁguration the electron occupying the valley state with energy E1,can-not decay to the other valley state at E GS due to Pauli spin blockade.It wouldﬁrst needs toﬂip its spin[28].We have estimatedτ↑↑andτ↑↓in our system(see supporting online text)andﬁnd thatτ↑↑>>h/k b T K>τ↑↓,where h/k b T K is the characteristic time-scale of the Kondo pro-cesses.Thus,the antiparallel spin conﬁguration will have relaxed before it has a change to build up a Kondo res-onance.Based on these lifetimes,we do not expect to observe the Kondo resonances associated with both an valley-state-and a spin-ﬂip.Finally,we investigate the degeneracy point of valley states in the Coulomb diamond of sample H64.This degeneracy point is indicated in Fig.1d by the red dashed ellipsoid.By means of the gate electrode,we can tune our system onto-or oﬀthis degeneracy point.The gate-tunability in this sample is created by a reconﬁguration of the level spectrum between the D0and D−-charge states,FIG.4:Colormap plot of I SD at V SD=0as a function of V G and B.For increasing B,a conductance peak develops around V G∼450mV at the valley degeneracy point(∆= 0),indicated by the dashed black line.Inset:Magneticﬁeld dependence of the valley degeneracy point.The resonance is ﬁxed at zero bias and its magnitude does not depend on the magneticﬁeld.probably due to Coulomb interactions in the D−-states. Figure4shows a colormap plot of I SD at V SD=0as a function of V G and B(at0.3K).Note that we are thus looking at the current associated with the central Kondo resonance.At B=0,we observe an increasing I SD for higher V G as the atom’s D−-level is pushed toward E F. As B is increased,the central Kondo resonance splits and moves away from V SD=0,see Fig.3.This leads to a general decrease in I SD.However,at around V G= 450mV a peak in I SD develops,indicated by the dashed black line.The applied B-ﬁeld splits oﬀthe resonances with spin-ﬂip,but it is the valley Kondo resonance here that stays at zero bias voltage giving rise to the local current peak.The inset of Fig.4shows the single Kondo resonance in dI SD/dV SD as a function of V SD and B.We observe that the magnitude of the resonance does not decrease signiﬁcantly with magneticﬁeld in contrast to the situation at∆=0(Fig.3b).This insensitivity of the Kondo eﬀect to magneticﬁeld which occurs only at∆= 0indicates the profound role of valley Kondo processes in our structure.It is noteworthy to mention that at this speciﬁc combination of V SD and V G the device can potentially work as a spin-ﬁlter[6].We acknowledge fruitful discussions with Yu.V. Nazarov,R.Joynt and S.Shiau.This project is sup-ported by the Dutch Foundation for Fundamental Re-search on Matter(FOM).6[1]Kondo,J.,Resistance Minimum in Dilute Magnetic Al-loys,Prog.Theor.Phys.3237-49(1964)[2]Hewson,A.C.,The Kondo Problem to Heavy Fermions(Cambridge Univ.Press,Cambridge,1993).[3]Wingreen N.S.,The Kondo eﬀect in novel systems,Mat.Science Eng.B842225(2001)and references therein.[4]Cox,D.L.,Zawadowski,A.,Exotic Kondo eﬀects in met-als:magnetic ions in a crystalline electricﬁeld and tun-neling centers,Adv.Phys.47,599-942(1998)[5]Inoshita,T.,Shimizu, A.,Kuramoto,Y.,Sakaki,H.,Correlated electron transport through a quantum dot: the multiple-level eﬀect.Phys.Rev.B48,14725-14728 (1993)[6]Borda,L.Zar´a nd,G.,Hofstetter,W.,Halperin,B.I.andvon Delft,J.,SU(4)Fermi Liquid State and Spin Filter-ing in a Double Quantum Dot System,Phys.Rev.Lett.90,026602(2003)[7]Zar´a nd,G.,Orbitalﬂuctuations and strong correlationsin quantum dots,Philosophical Magazine,86,2043-2072 (2006)[8]Jarillo-Herrero,P.,Kong,J.,van der Zant H.S.J.,Dekker,C.,Kouwenhoven,L.P.,De Franceschi,S.,Or-bital Kondo eﬀect in carbon nanotubes,Nature434,484 (2005)[9]Martins,A.S.,Capaz,R.B.and Koiller,B.,Electric-ﬁeldcontrol and adiabatic evolution of shallow donor impuri-ties in silicon,Phys.Rev.B69,085320(2004)[10]Lansbergen,G.P.et al.,Gate induced quantum conﬁne-ment transition of a single dopant atom in a Si FinFET, Nature Physics4,656(2008)[11]Rokhinson,L.P.,Guo,L.J.,Chou,S.Y.,Tsui, D.C.,Kondo-like zero-bias anomaly in electronic transport through an ultrasmall Si quantum dot,Phys.Rev.B60, R16319-R16321(1999)[12]Specht,M.,Sanquer,M.,Deleonibus,S.,Gullegan G.,Signature of Kondo eﬀect in silicon quantum dots,Eur.Phys.J.B26,503-508(2002)[13]Klein,L.J.,Savage, D.E.,Eriksson,M.A.,Coulombblockade and Kondo eﬀect in a few-electron silicon/silicon-germanium quantum dot,Appl.Phys.Lett.90,033103(2007)[14]Shiau,S.,Chutia,S.and Joynt,R.,Valley Kondo eﬀectin silicon quantum dots,Phys.Rev.B75,195345(2007) [15]Roch,N.,Florens,S.,Bouchiat,V.,Wernsdirfer,W.,Balestro, F.,Quantum phase transistion in a single molecule quantum dot,Nature453,633(2008)[16]Sellier,H.et al.,Transport Spectroscopy of a SingleDopant in a Gated Silicon Nanowire,Phys.Rev.Lett.97,206805(2006)[17]Calvet,L.E.,Wheeler,R.G.and Reed,M.A.,Observa-tion of the Linear Stark Eﬀect in a Single Acceptor in Si, Phys.Rev.Lett.98,096805(2007)[18]Hofheinz,M.et al.,Individual charge traps in siliconnanowires,Eur.Phys.J.B54,299307(2006)[19]Pierre,M.,Hofheinz,M.,Jehl,X.,Sanquer,M.,Molas,G.,Vinet,M.,Deleonibus S.,Oﬀset charges acting as ex-cited states in quantum dots spectroscopy,Eur.Phys.J.B70,475-481(2009)[20]Goldhaber-Gordon,D.,Gres,J.,Kastner,M.A.,Shtrik-man,H.,Mahalu, D.,Meirav,U.,From the Kondo Regime to the Mixed-Valence Regime in a Single-Electron Transistor,Phys.Rev.Lett.81,5225(1998) [21]Although the value of s=0.22stems from SU(2)spinKondo processes,it is valid for SU(4)-Kondo systems as well[8,25].[22]Paaske,J.,Rosch,A.,W¨o lﬂe,P.,Mason,N.,Marcus,C.M.,Nyg˙ard,Non-equilibrium singlet-triplet Kondo ef-fect in carbon nanotubes,Nature Physics2,460(2006) [23]Osorio, E.A.et al.,Electronic Excitations of a SingleMolecule Contacted in a Three-Terminal Conﬁguration, Nanoletters7,3336-3342(2007)[24]Meir,Y.,Wingreen,N.S.,Lee,P.A.,Low-TemperatureTransport Through a Quantum Dot:The Anderson Model Out of Equilibrium,Phys.Rev.Lett.70,2601 (1993)[25]Lim,J.S.,Choi,M-S,Choi,M.Y.,L´o pez,R.,Aguado,R.,Kondo eﬀects in carbon nanotubes:From SU(4)to SU(2)symmetry,Phys.Rev.B74,205119(2006) [26]Hada,Y.,Eto,M.,Electronic states in silicon quan-tum dots:Multivalley artiﬁcial atoms,Phys.Rev.B68, 155322(2003)[27]Eto,M.,Hada,Y.,Kondo Eﬀect in Silicon QuantumDots with Valley Degeneracy,AIP Conf.Proc.850,1382-1383(2006)[28]A comparable process in the direct transport throughSi/SiGe double dots(Lifetime Enhanced Transport)has been recently proposed[29].[29]Shaji,N.et.al.,Spin blockade and lifetime-enhancedtransport in a few-electron Si/SiGe double quantum dot, Nature Physics4,540(2008)7Supporting InformationFinFET DevicesThe FinFETs used in this study consist of a silicon nanowire connected to large contacts etched in a60nm layer of p-type Silicon On Insulator.The wire is covered with a nitrided oxide(1.4nm equivalent SiO2thickness) and a narrow poly-crystalline silicon wire is deposited perpendicularly on top to form a gate on three faces.Ion implantation over the entire surface forms n-type degen-erate source,drain,and gate electrodes while the channel protected by the gate remains p-type,see Fig.1a of the main article.The conventional operation of this n-p-n ﬁeld eﬀect transistor is to apply a positive gate voltage to create an inversion in the channel and allow a current toﬂow.Unintentionally,there are As donors present be-low the Si/SiO2interface that show up in the transport characteristics[1].Relation between∆and T KThe information obtained on T K in the main article allows us to investigate the relation between the splitting (∆)of the ground(E GS)-andﬁrst excited(E1)-state and T K.It is expected that T K decreases as∆increases, since a high∆freezes out valley-stateﬂuctuations.The relationship between T K of an SU(4)system and∆was calculated by Eto[2]in a poor mans scaling approach ask B T K(∆) B K =k B T K(∆=0)ϕ(2)whereϕ=ΓE1/ΓGS,withΓE1andΓGS the lifetimes of E1and E GS respectively.Due to the small∆com-pared to the barrier height between the atom and the source/drain contact,we expectϕ∼1.Together with ∆=1meV and T K∼2.7K(for sample H67)and∆=2meV and T K∼6K(for sample J17),Eq.2yields k B T K(∆)/k B T K(∆=0)=0.4and k B T K(∆)/k B T K(∆= 0)=0.3respectively.We can thus conclude that the rela-tively high∆,which separates E GS and E1well in energy, will certainly quench valley-stateﬂuctuations to a certain degree but is not expected to reduce T K to a level that Valley eﬀects become obscured.Valley Kondo density of statesHere,we explain in some more detail the relation be-tween the density of states induced by the Kondo eﬀects and the resulting current.The Kondo density of states (DOS)has three main peaks,see Fig.1a.A central peak at E F=0due to processes without valley-stateﬂip and two peaks at E F=±∆due to processes with valley-state ﬂip,as explained in the main text.Even a small asym-metry(ϕclose to1)will split the Valley Kondo DOS into an SU(2)-and an SU(4)-part[3],indicated in Fig1b in black and red respectively.The SU(2)-part is positioned at E F=0or E F=±∆,while the SU(4)-part will be shifted to slightly higher positive energy(on the order of T K).A voltage bias applied between the source and FIG.1:(a)dI SD/dV SD as a function of V SD in the Kondo regime(at395mV G)of sample J17.The substructure in the Kondo resonances is the result of a small diﬀerence between ΓE1andΓGS.This splits the peaks into a(central)SU(2)-part (black arrows)and two SU(4)-peaks(red arrows).(b)Density of states in the channel as a result ofϕ(=ΓE1/ΓGS)<1and applied V SD.drain leads results in the Kondo peaks to split,leaving a copy of the original structure in the DOS now at the E F of each lead,which is schematically indicated in Fig.1b by a separate DOS associated with each contact.The current density depends directly on the density of states present within the bias window deﬁned by source/drain (indicated by the gray area in Fig1b)[4].The splitting between SU(2)-and SU(4)-processes will thus lead to a three-peak structure as a function of V SD.Figure.1a has a few more noteworthy features.The zero-bias resonance is not positioned exactly at V SD=0, as can also be observed in the transport data(Fig1c of the main article)where it is a few hundredµeV above the Fermi energy near the D0charge state and a few hundredµeV below the Fermi energy near the D−charge state.This feature is also known to arise in the Kondo strong coupling limit[5,6].We further observe that the resonances at V SD=+/-2mV diﬀer substantially in magnitude.This asymmetry between the two side-peaks can actually be expected from SU(4)Kondo sys-tems where∆is of the same order as(but of course al-ways smaller than)the energy spacing between E GS and。

湖师大文献检索作业

医学文献检索作业姓名：学号：医院：第一次作业一、在期刊导航中查找：1、与你专业相关的期刊有7种。

按照期刊的综合影响因子排名，请填写排名前三的期刊的相关信息？包括刊名、主办单位、2、CNKI中收录的中华医学会创办的期刊有155种。

按照期刊的被引次数排名，请填写排名前三的期刊的相关信息？包括刊名、综合影响因子、3.请查找医药卫生类核心期刊有252种，其中内科学类有24种。

4.自学作业：（提示：在百度上查找）（1）核心期刊的含义以及中国四大核心期刊体系。

核心期：刊是期刊中学术水平较高的刊物，是进行刊物评价而非具体学术评价的工具。

中国四大核心期刊体系：①中国科学技术信息研究所“中国科技论文统计源期刊”(又称“中国科技核心期刊”)：②北京大学图书馆“中文核心期刊”；③南京大学“中文社会科学引文索引(CSSCI)来源期刊”；④中国社会科学院文献信息中心“中国人文社会科学核心期刊”。

（2）期刊的影响因子（IF）的定义和计算公式。

①期刊的影响因子(Impact factor，IF)，是代表期刊影响大小的一项定量指标。

也就是某刊平均每篇论文的被引用数，它实际上是某刊在某年被全部源刊物引证该刊前两年发表论文的次数，与该刊前两年所发表的全部源论文数之比。

②计算公式：IF（k）= (nk-1+nk-2) / ( Nk-1+Nk-2)说明：k 为某年, Nk-1+Nk-2 为该刊在前一两年发表的论文数量, nk-1 和nk-2 该刊在k 年的被引用数量。

也就是说，某刊在2005年的影响因子是其2004和2003两年刊载的论文在2005年的被引总数除该刊在2004和2003这两年的载文总数（可引论文）。

（3）计算：某期刊2015年影响因子本刊2014年的文章在2015年的被引次数：48 ；本刊2014年的发文量：187本刊2013年的文章在2015年的被引次数：128 ；本刊2013年的发文量：154计算过程及结果：（48+128）/（187+154）=0.5165.（1）以“湖南师范大学医学院”为机构名检索2010~2016发表的论文数量，其中在核心期刊上发表的论文数量，在核心期刊上发文最多的作者；“湖南师范大学医学院”为机构名检索2010~2016发表的论文数量为：797篇；其中在核心期刊上发表的论文数量为：198篇；在核心期刊上发文最多的作者是：曹建国（13篇）。

Quantum tomography of the GHZ state

1
Quantum tomography of the GHZ state
by
G. M. D’Arianoa , M. Rubinb , M. F. Sacchia , and Y. Shihb ,
arXiv:quant-ph/9906067v1 18 Jun 1999
a
Dipartimento di Fisica ‘A. Volta’, Universit` a di Pavia and INFM Unit` a di Pavia via A. Bassi 6, I-27100 Pavia, Italy b Physics Department, University of Maryland, Baltimore County, Baltimore, Maryland 21228 Abstract. We present a method of generation of the Greenberger–Horne– Zeilinger state involving type II and type I parametric downconversion, and triggering photodetectors. The state generated by the proposed experimental set-up can be reconstructed through multi-mode quantum homodyne tomography. The feasibility of the measurement is studied on the basis of Monte-Carlo simulations.
1 Introduction
A number of proposals for generating the Greenberger–Horne–Zeilinger (GHZ) state [1] has been suggested in the literature [2]. Such kind of state is very interesting as it leads to correlations between three particles in contradiction with the EinsteinPodolsky-Rosen idea of “elements of reality” [3]. In the present contribution we present a scheme for a complete quantum test of a GHZ state of radiation, not just for a simple veriﬁcation of some GHZ correlations, which do not prove that a true GHZ state has been produced. In fact, the veriﬁcation of a state-preparation procedure needs a complete state-reconstruction technique, whereas correlation measurements [4] give identical results for very diﬀerent states of radiation. In this respect, a crucial technique for state-preparation tests is quantum homodyne tomography, in which the detrimental eﬀect of non-unity quantum eﬃciency of detectors is taken into account ab initio by the reconstruction algorithms. In the following we propose a method for generating a GHZ state through type II and type I parametric downconversion, and triggering photodetectors. The proposed set-up, although it has low rate of production due to low eﬃciency for single-photon downconversion, however is the only way to generate a “true” GHZ state, without an additional vacuum component. The scheme allows a tomographic state-reconstruction, whose feasibility here is studied on the basis of Monte–Cacheme for the GHZ-state generation

多光子态的干涉和区分

附件2论文中英文摘要作者姓名：孙方稳论文题目：多光子态的干涉和区分作者简介：孙方稳，男，1979年10月出生，2004年9月师从于中国科学技术大学郭光灿教授，于2007年6月获博士学位。

中文摘要光学，作为一种工具，在物理学的基础研究和各种应用领域中都发挥着巨大的作用。

特别是在激光被发明后，很多重要的研究和发现都是以它为基础完成的。

而光的相干性研究又是光学应用的基础，对于研究物理学基础领域例如各种物质与光的相互作用有重要的用处；在应用领域，光的相干性研究对于高精密测量、图象处理等都有重要的推动作用。

此外，在新兴的量子信息技术中，光的相干性研究也起到了重要的推动作用。

在量子信息技术中, 量子力学的量子态叠加理论和测量塌缩理论保证了量子密钥分配的安全；同样量子态叠加理论和演化理论使得量子计算可以并行工作，其效率将远优于普通电子计算机。

在实验上，人们尝试着在多种物理体系上实现量子通讯和量子计算。

而光学方法因为其独特的优点，较其他方法发展更快。

首先，光子的易传输性是其他物理体系不具备的。

因此在实际量子密钥分配中，几乎都使用光子作为信息载体。

其次，成熟的非线性光学手段和良好的激光光源使得光子态制备变的容易。

最后，对于光子态的精密控制和探测也使得人们容易完成不同的量子操控方案。

所以，诸如量子密钥分配，量子力学非局域性检验，量子隐形传态等一系列实验都是首先利用光子完成的。

实际上，用在量子信息中的光子相干特性在物理本质上就是某种光子干涉，都是建立在各种干涉模型上的。

如实际的量子密钥分配的光学部分就是一种单光子干涉；而利用线性光学方法来完成的量子计算方案是多光子干涉。

因此对光子干涉本身的研究将为其在量子信息中的应用提供更好的方法。

反过来，光子干涉也是一种量子操作过程。

我们可以利用量子信息的语言来描述和分析干涉。

例如光子干涉时对态的转换可以表示成量子信息中的量子比特操作；而其干涉可见度的不同就是量子信息中的消相干作用的结果。

量子信息——连续变量篇

Quantum information with continuous variablesSamuel L.BraunsteinComputer Science,University of York,York YO105DD,United KingdomPeter van LoockNational Institute of Informatics(NII),Tokyo101-8430,Japan and Institute of TheoreticalPhysics,Institute of Optics,Information and Photonics(Max-Planck Forschungsgruppe),Universität Erlangen-Nürnberg,D-91058Erlangen,Germany͑Published29June2005͒Quantum information is a rapidly advancing area of interdisciplinary research.It may lead to real-world applications for communication and computation unavailable without the exploitation of quantum properties such as nonorthogonality or entanglement.This article reviews the progress in quantum information based on continuous quantum variables,with emphasis on quantum optical implementations in terms of the quadrature amplitudes of the electromagneticﬁeld.CONTENTSI.Introduction513II.Continuous Variables in Quantum Optics516A.The quadratures of the quantizedﬁeld516B.Phase-space representations518C.Gaussian states519D.Linear optics519E.Nonlinear optics520F.Polarization and spin representations522G.Necessity of phase reference523 III.Continuous-Variable Entanglement523A.Bipartite entanglement5251.Pure states5252.Mixed states and inseparability criteria526B.Multipartite entanglement5291.Discrete variables5292.Genuine multipartite entanglement5303.Separability properties of Gaussian states5304.Generating entanglement5315.Measuring entanglement533C.Bound entanglement534D.Nonlocality5341.Traditional EPR-type approach5352.Phase-space approach5363.Pseudospin approach536E.Verifying entanglement experimentally537 IV.Quantum Communication with Continuous Variables538A.Quantum teleportation5401.Teleportation protocol5412.Teleportation criteria5433.Entanglement swapping546B.Dense coding546rmation:A measure5472.Mutual information5473.Classical communication5474.Classical communication via quantum states5475.Dense coding548C.Quantum error correction550D.Quantum cryptography5501.Entanglement-based versus prepare andmeasure5502.Early ideas and recent progress5513.Absolute theoretical security5524.Verifying experimental security5535.Quantum secret sharing553E.Entanglement distillation554F.Quantum memory555V.Quantum Cloning with Continuous Variables555A.Local universal cloning5551.Beyond no-cloning5552.Universal cloners556B.Local cloning of Gaussian states5571.Fidelity bounds for Gaussian cloners5572.An optical cloning circuit for coherentstates558C.Telecloning559 VI.Quantum Computation with Continuous Variables560A.Universal quantum computation560B.Extension of the Gottesman-Knill theorem563 VII.Experiments with Continuous Quantum Variables565A.Generation of squeezed-state EPR entanglement5651.Broadband entanglement via opticalparametric ampliﬁcation5652.Kerr effect and linear interference567B.Generation of long-lived atomic entanglement568C.Generation of genuine multipartite entanglement569D.Quantum teleportation of coherent states569E.Experimental dense coding570F.Experimental quantum key distribution571G.Demonstration of a quantum memory effect572 VIII.Concluding Remarks572 Acknowledgments573 References573I.INTRODUCTIONQuantum information is a relatively young branch of physics.One of its goals is to interpret the concepts of quantum physics from an information-theoretic point of view.This may lead to a deeper understanding of quan-REVIEWS OF MODERN PHYSICS,VOLUME77,APRIL20050034-6861/2005/77͑2͒/513͑65͒/$50.00©2005The American Physical Society513tum theory.Conversely,information and computation are intrinsically physical concepts,since they rely on physical systems in which information is stored and by means of which information is processed or transmitted. Hence physical concepts,and at a more fundamental level quantum physical concepts,must be incorporated in a theory of information and computation.Further-more,the exploitation of quantum effects may even prove beneﬁcial for various kinds of information pro-cessing and communication.The most prominent ex-amples of this are quantum computation and quantum key distribution.Quantum computation means in par-ticular cases,in principle,computation faster than any known classical computation.Quantum key distribution makes possible,in principle,unconditionally secure communication as opposed to communication based on classical key distribution.From a conceptual point of view,it is illuminating to consider continuous quantum variables in quantum in-formation theory.This includes the extension of quan-tum communication protocols from discrete to continu-ous variables and hence fromﬁnite to inﬁnite dimensions.For instance,the original discrete-variable quantum teleportation protocol for qubits and other ﬁnite-dimensional systems͑Bennett et al.,1993͒was soon after its publication translated into the continuous-variable setting͑Vaidman,1994͒.The main motivation for dealing with continuous variables in quantum infor-mation,however,originated in a more practical observa-tion:efﬁcient implementation of the essential steps in quantum communication protocols,namely,preparing, unitarily manipulating,and measuring͑entangled͒quan-tum states,is achievable in quantum optics utilizing con-tinuous quadrature amplitudes of the quantized electro-magneticﬁeld.For example,the tools for measuring a quadrature with near-unit efﬁciency or for displacing an optical mode in phase space are provided by homodyne-detection and feedforward techniques,respectively. Continuous-variable entanglement can be efﬁciently produced using squeezed light͓in which the squeezing of a quadrature’s quantumﬂuctuations is due to a non-linear optical interaction͑Walls and Milburn,1994͔͒and linear optics.A valuable feature of quantum optical implementa-tions based upon continuous variables,related to their high efﬁciency,is their unconditionalness.Quantum re-sources such as entangled states emerge from the non-linear optical interaction of a laser with a crystal͑supple-mented if necessary by some linear optics͒in an unconditional fashion,i.e.,every inverse bandwidth time.This unconditionalness is hard to obtain in discrete-variable qubit-based implementations using single-photon states.In that case,the desired prepara-tion due to the nonlinear optical interaction depends on particular͑coincidence͒measurement results ruling out the unwanted͑in particular,vacuum͒contributions in the outgoing state vector.However,the unconditional-ness of the continuous-variable implementations has its price:it is at the expense of the quality of the entangle-ment of the prepared states.This entanglement and hence any entanglement-based quantum protocol is al-ways imperfect,the degree of imperfection depending on the amount of squeezing of the laser light involved. Good quality and performance require large squeezing which is technologically demanding,but to a certain ex-tent͓about10dB͑Wu et al.,1986͔͒already state of the art.Of course,in continuous-variable protocols that do not rely on entanglement,for instance,coherent-state-based quantum key distribution,these imperfections do not occur.To summarize,in the most commonly used optical ap-proaches,the continuous-variable implementations al-ways work pretty well͑and hence efﬁciently and uncon-ditionally͒,but never perfectly.Their discrete-variable counterparts only work sometimes͑conditioned upon rare successful events͒,but they succeed,in principle, perfectly.A similar tradeoff occurs when optical quan-tum states are sent through noisy channels͑opticalﬁ-bers͒,for example,in a realistic quantum key distribu-tion scenario.Subject to losses,the continuous-variable states accumulate noise and emerge at the receiver as contaminated versions of the sender’s input states.The discrete-variable quantum information encoded in single-photon states is reliably conveyed for each photon that is not absorbed during transmission.Due to the recent results of Knill,Laﬂamme,and Mil-burn͑Knill et al.,2001͒,it is now known that efﬁcient quantum information processing is possible,in principle, solely by means of linear optics.Their scheme is formu-lated in a discrete-variable setting in which the quantum information is encoded in single-photon states.Apart from entangled auxiliary photon states,generated off-line without restriction to linear optics,conditional dy-namics͑feedforward͒is the essential ingredient in mak-ing this approach work.Universal quantum gates such as a controlled-NOT gate can,in principle,be built using this scheme without need of any Kerr-type nonlinear op-tical interaction͑corresponding to an interaction Hamil-tonian quartic in the optical modes’annihilation and creation operators͒.This Kerr-type interaction would be hard to obtain on the level of single photons.However, the off-line generation of the complicated auxiliary states needed in the Knill-Laﬂamme-Milburn scheme seems impractical too.Similarly,in the continuous-variable setting,when it comes to more advanced quantum information proto-cols,such as universal quantum computation or,in a communication scenario,entanglement distillation,it turns out that tools more sophisticated than mere Gaussian operations are needed.In fact,the Gaussian operations are effectively those described by interaction Hamiltonians at most quadratic in the optical modes’annihilation and creation operators,thus leading to lin-ear input-output relations as in beam-splitter or squeez-ing transformations.Gaussian operations,mapping Gaussian states onto Gaussian states,also include ho-modyne detections and phase-space displacements.In contrast,the non-Gaussian operations required for ad-vanced continuous-variable quantum communication͑in particular,long-distance communication based on en-514S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005tanglement distillation and swapping,quantum memory,and teleportation͒are due either to at least cubic non-linear optical interactions or to conditional transforma-tions depending on non-Gaussian measurements such asphoton counting.It seems that,at this very sophisticatedlevel,the difﬁculties and requirements of the discrete-and continuous-variable implementations are analogous.In this review,our aim is to highlight the strengths ofthe continuous-variable approaches to quantum infor-mation processing.Therefore we focus on those proto-cols that are based on Gaussian states and their feasiblemanipulation through Gaussian operations.This leads tocontinuous-variable proposals for the implementation ofthe simplest quantum communication protocols,such asquantum teleportation and quantum key distribution,and includes the efﬁcient generation and detection ofcontinuous-variable entanglement.Before dealing with quantum communication andcomputation,in Sec.II,weﬁrst introduce continuousquantum variables within the framework of quantumoptics.The discussions about the quadratures of quan-tized electromagnetic modes,about phase-space repre-sentations,and about Gaussian states include the nota-tions and conventions that we use throughout thisarticle.We conclude Sec.II with a few remarks on linearand nonlinear optics,on alternative polarization andspin representations,and on the necessity of a phasereference in continuous-variable implementations.Thenotion of entanglement,indispensable in many quantumprotocols,is described in Sec.III in the context of con-tinuous variables.We discuss pure and mixed entangledstates,entanglement between two͑bipartite͒and be-tween many͑multipartite͒parties,and so-called bound ͑undistillable͒entanglement.The generation,measure-ment,and veriﬁcation͑both theoretical and experimen-tal͒of continuous-variable entanglement are here of par-ticular interest.As for the properties of the continuous-variable entangled states related with theirinseparability,we explain how the nonlocal character ofthese states is revealed.This involves,for instance,vio-lations of Bell-type inequalities imposed by local real-ism.Such violations,however,cannot occur when themeasurements considered are exclusively of continuous-variable type.This is due to the strict positivity of theWigner function of the Gaussian continuous-variable en-tangled states,which allows for a hidden-variable de-scription in terms of the quadrature observables.In Sec.IV,we describe the conceptually and practi-cally most important quantum communication protocols formulated in terms of continuous variables and thus utilizing the continuous-variable͑entangled͒states. These schemes include quantum teleportation and en-tanglement swapping͑teleportation of entanglement͒, quantum͑super͒dense coding,quantum error correc-tion,quantum cryptography,and entanglement distilla-tion.Since quantum teleportation based on nonmaxi-mum continuous-variable entanglement,usingﬁnitely squeezed two-mode squeezed states,is always imperfect, teleportation criteria are needed both for the theoretical and for the experimental veriﬁcation.As is known from classical communication,light,propagating at high speed and offering a broad range of different frequen-cies,is an ideal carrier for the transmission of informa-tion.This applies to quantum communication as well. However,light is less suited for the storage of informa-tion.In order to store quantum information,for in-stance,at the intermediate stations in a quantum re-peater,atoms are more appropriate media than light. Signiﬁcantly,as another motivation to deal with continu-ous variables,a feasible light-atom interface can be built via free-space interaction of light with an atomic en-semble based on the alternative polarization and spin-type variables.No strong cavity QED coupling is needed as with single photons.The concepts of this transfer of quantum information from light to atoms and vice versa, as the essential ingredients of a quantum memory,are discussed in Sec.IV.FSection V is devoted to quantum cloning with con-tinuous variables.One of the most fundamental͑and historically one of theﬁrst͒“laws”of quantum informa-tion theory is the so-called no-cloning theorem͑Dieks, 1982;Wootters and Zurek,1982͒.It forbids the exact copying of arbitrary quantum states.However,arbitrary quantum states can be copied approximately,and the resemblance͑in mathematical terms,the overlap orﬁ-delity͒between the clones may attain an optimal value independent of the original states.Such optimal cloning can be accomplished locally by sending the original states͑together with some auxiliary system͒through a local unitary quantum circuit.Optimal cloning of Gauss-ian continuous-variable states appears to be more inter-esting than that of general continuous-variable states, because the latter can be mimicked by a simple coin toss.We describe a non-entanglement-based implemen-tation for the optimal local cloning of Gaussian continuous-variable states.In addition,for Gaussian continuous-variable states,an optical implementation exists of optimal cloning at a distance͑telecloning͒.In this case,the optimality requires entanglement.The cor-responding multiparty entanglement is again producible with nonlinear optics͑squeezed light͒and linear optics ͑beam splitters͒.Quantum computation over continuous variables,dis-cussed in Sec.VI,is a more subtle issue than the in some sense straightforward continuous-variable extensions of quantum communication protocols.Atﬁrst sight,con-tinuous variables do not appear well suited for the pro-cessing of digital information in a computation.On the other hand,a continuous-variable quantum state having an inﬁnite-dimensional spectrum of eigenstates contains a vast amount of quantum information.Hence it might be promising to adjust the continuous-variable states theoretically to the task of computation͑for instance,by discretization͒and yet to exploit their continuous-variable character experimentally in efﬁcient͑optical͒implementations.We explain in Sec.VI why universal quantum computation over continuous variables re-quires Hamiltonians at least cubic in the position and momentum͑quadrature͒operators.Similarly,any quan-tum circuit that consists exclusively of unitary gates from515S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005the continuous-variable Clifford group can be efﬁcientlysimulated by purely classical means.This is acontinuous-variable extension of the discrete-variableGottesman-Knill theorem in which the Clifford groupelements include gates such as the Hadamard͑in thecontinuous-variable case,Fourier͒transform or the con-trolled NOT͑CNOT͒.The theorem applies,for example,to quantum teleportation which is fully describable by CNOT’s and Hadamard͑or Fourier͒transforms of some eigenstates supplemented by measurements in thateigenbasis and spin or phaseﬂip operations͑or phase-space displacements͒.Before some concluding remarks in Sec.VIII,wepresent some of the experimental approaches to squeez-ing of light and squeezed-state entanglement generationin Sec.VII.A.Both quadratic and quartic optical nonlin-earities are suitable for this,namely,parametric downconversion and the Kerr effect,respectively.Quantumteleportation experiments that have been performed al-ready based on continuous-variable squeezed-state en-tanglement are described in Sec.VII.D.In Sec.VII,wefurther discuss experiments with long-lived atomic en-tanglement,with genuine multipartite entanglement ofoptical modes,experimental dense coding,experimentalquantum key distribution,and the demonstration of aquantum memory effect.II.CONTINUOUS VARIABLES IN QUANTUM OPTICSFor the transition from classical to quantum mechan-ics,the position and momentum observables of the par-ticles turn into noncommuting Hermitian operators inthe Hamiltonian.In quantum optics,the quantized elec-tromagnetic modes correspond to quantum harmonicoscillators.The modes’quadratures play the roles of theoscillators’position and momentum operators obeyingan analogous Heisenberg uncertainty relation.A.The quadratures of the quantizedﬁeldFrom the Hamiltonian of a quantum harmonic oscil-lator expressed in terms of͑dimensionless͒creation and annihilation operators and representing a single mode k, Hˆk=ប␻k͑aˆk†aˆk+12͒,we obtain the well-known form writ-ten in terms of“position”and“momentum”operators ͑unit mass͒,Hˆk=12͑pˆk2+␻k2xˆk2͒,͑1͒withaˆk=1ͱ2ប␻k͑␻k xˆk+ipˆk͒,͑2͒aˆk†=1ͱ2ប␻k͑␻k xˆk−ipˆk͒,͑3͒or,conversely,xˆk=ͱប2␻k͑aˆk+aˆk†͒,͑4͒pˆk=−iͱប␻k2͑aˆk−aˆk†͒.͑5͒Here,we have used the well-known commutation rela-tion for position and momentum,͓xˆk,pˆkЈ͔=iប␦kkЈ,͑6͒which is consistent with the bosonic commutation rela-tions͓aˆk,aˆkЈ†͔=␦kkЈ,͓aˆk,aˆkЈ͔=0.In Eq.͑2͒,we see that up to normalization factors the position and the momentum are the real and imaginary parts of the annihilation op-erator.Let us now deﬁne the dimensionless pair of con-jugate variables,Xˆkϵͱ␻k2បxˆk=Re aˆk,Pˆkϵ1ͱ2ប␻k pˆk=Im aˆk.͑7͒Their commutation relation is then͓Xˆk,PˆkЈ͔=i2␦kkЈ.͑8͒In other words,the dimensionless position and momen-tum operators,Xˆk and Pˆk,are deﬁned as if we setប=1/2.These operators represent the quadratures of a single mode k,in classical terms corresponding to the real and imaginary parts of the oscillator’s complex am-plitude.In the following,by using͑Xˆ,Pˆ͒or equivalently ͑xˆ,pˆ͒,we shall always refer to these dimensionless quadratures as playing the roles of position and momen-tum.Hence͑xˆ,pˆ͒will also stand for a conjugate pair of dimensionless quadratures.The Heisenberg uncertainty relation,expressed in terms of the variances of two arbitrary noncommuting observables Aˆand Bˆfor an arbitrary given quantum state,͗͑⌬Aˆ͒2͘ϵŠ͑Aˆ−͗Aˆ͒͘2‹=͗Aˆ2͘−͗Aˆ͘2,͗͑⌬Bˆ͒2͘ϵŠ͑Bˆ−͗Bˆ͒͘2‹=͗Bˆ2͘−͗Bˆ͘2,͑9͒becomes͗͑⌬Aˆ͒2͗͑͘⌬Bˆ͒2͘ജ14͉͓͗Aˆ,Bˆ͔͉͘2.͑10͒Inserting Eq.͑8͒into Eq.͑10͒yields the uncertainty re-lation for a pair of conjugate quadrature observables of a single mode k,xˆk=͑aˆk+aˆk†͒/2,pˆk=͑aˆk−aˆk†͒/2i,͑11͒namely,͗͑⌬xˆk͒2͗͑͘⌬pˆk͒2͘ജ14͉͓͗xˆk,pˆk͔͉͘2=116.͑12͒Thus,in our units,the quadrature variance for a vacuum or coherent state of a single mode is1/4.Let us further516S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005illuminate the meaning of the quadratures by looking at a single frequency mode of the electric ﬁeld ͑for a single polarization ͒,E ˆk ͑r ,t ͒=E 0͓a ˆk ei ͑k ·r −␻k t ͒+a ˆk †e −i ͑k ·r −␻k t ͔͒.͑13͒The constant E 0contains all the dimensional prefactors.By using Eq.͑11͒,we can rewrite the mode asE ˆk ͑r ,t ͒=2E 0͓x ˆk cos ͑␻k t −k ·r ͒+pˆk sin ͑␻k t −k ·r ͔͒.͑14͒Clearly,the position and momentum operators xˆk and p ˆk represent the in-phase and out-of-phase components of the electric-ﬁeld amplitude of the single mode k with respect to a ͑classical ͒reference wave ϰcos ͑␻k t −k ·r ͒.The choice of the phase of this wave is arbitrary,of course,and a more general reference wave would lead us to the single-mode descriptionE ˆk ͑r ,t ͒=2E 0͓x ˆk ͑⌰͒cos ͑␻k t −k ·r −⌰͒+pˆk ͑⌰͒sin ͑␻k t −k ·r −⌰͔͒,͑15͒with the more general quadraturesxˆk ͑⌰͒=͑a ˆk e −i ⌰+a ˆk †e +i ⌰͒/2,͑16͒p ˆk ͑⌰͒=͑a ˆk e −i ⌰−a ˆk †e +i ⌰͒/2i .͑17͒These new quadratures can be obtained from x ˆk and p ˆk via the rotationͩx ˆk ͑⌰͒pˆk ͑⌰͒ͪ=ͩcos ⌰sin ⌰−sin ⌰cos ⌰ͪͩxˆk pˆk ͪ.͑18͒Since this is a unitary transformation,we again end upwith a pair of conjugate observables fulﬁlling the com-mutation relation ͑8͒.Furthermore,because pˆk ͑⌰͒=x ˆk ͑⌰+␲/2͒,the whole continuum of quadratures is cov-ered by x ˆk ͑⌰͒with ⌰෈͓0,␲͒.This continuum of observ-ables is indeed measurable by relatively simple means.Such a so-called homodyne detection works as follows.A photodetector measuring an electromagnetic mode converts the photons into electrons and hence into an electric current,called the photocurrent i ˆ.It is therefore sensible to assume i ˆϰn ˆ=a ˆ†a ˆor i ˆ=qaˆ†a ˆwhere q is a con-stant ͑Paul,1995͒.In order to detect a quadrature of themode aˆ,the mode must be combined with an intense local oscillator at a 50:50beam splitter.The local oscil-lator is assumed to be in a coherent state with large photon number,͉␣LO ͘.It is therefore reasonable to de-scribe this oscillator by a classical complex amplitude␣LO rather than by an annihilation operator aˆLO .The two output modes of the beam splitter,͑aˆLO +a ˆ͒/ͱ2and ͑a ˆLO −a ˆ͒/ͱ2͑see Sec.II.D ͒,may then be approximated byaˆ1=͑␣LO +a ˆ͒/ͱ2,aˆ2=͑␣LO −a ˆ͒/ͱ2.͑19͒This yields the photocurrentsi ˆ1=qa ˆ1†aˆ1=q ͑␣LO *+a ˆ†͒͑␣LO +a ˆ͒/2,i ˆ2=qa ˆ2†aˆ2=q ͑␣LO *−a ˆ†͒͑␣LO −a ˆ͒/2.͑20͒The actual quantity to be measured will be the differ-ence photocurrent␦i ˆϵi ˆ1−i ˆ2=q ͑␣LO *aˆ+␣LO a ˆ†͒.͑21͒By introducing the phase ⌰of the local oscillator,␣LO=͉␣LO ͉exp ͑i ⌰͒,we recognize that the quadrature observ-able xˆ͑⌰͒from Eq.͑16͒is measured ͑without mode index k ͒.Now adjustment of the local oscillator’s phase ⌰෈͓0,␲͔enables us to detect any quadrature from thewhole continuum of quadratures xˆ͑⌰͒.A possible way to realize quantum tomography ͑Leonhardt,1997͒,i.e.,the reconstruction of the mode’s quantum state given by its Wigner function,relies on this measurement method,called ͑balanced ͒homodyne detection .A broadband rather than a single-mode description of homodyne de-tection can be found in the work of Braunstein and Crouch ͑1991͒,who also investigate the inﬂuence of a quantized local oscillator.We have now seen that it is not too hard to measure the quadratures of an electromagnetic mode.Unitary transformations such as quadrature displacements ͑phase-space displacements ͒can also be relatively easily performed via the so-called feedforward technique,as opposed to,for example,photon number displacements.This simplicity and the high efﬁciency when measuring and manipulating continuous quadratures are the main reasons why continuous-variable schemes appear more attractive than those based on discrete variables such as the photon number.In the following,we shall refer mainly to the conju-gate pair of quadratures xˆk and p ˆk ͑position and momen-tum,i.e.,⌰=0and ⌰=␲/2͒.In terms of these quadra-tures,the number operator becomesn ˆk =a ˆk †a ˆk =x ˆk 2+p ˆk 2−12,͑22͒using Eq.͑8͒.Let us ﬁnally review some useful formulas for the single-mode quadrature eigenstates,xˆ͉x ͘=x ͉x ͘,pˆ͉p ͘=p ͉p ͘,͑23͒where we have now dropped the mode index k .They are orthogonal,͗x ͉x Ј͘=␦͑x −x Ј͒,͗p ͉p Ј͘=␦͑p −p Ј͒,͑24͒and complete,͵−ϱϱ͉x ͗͘x ͉dx =1,͵−ϱϱ͉p ͗͘p ͉dp =1.͑25͒Just as for position and momentum eigenstates,the quadrature eigenstates are mutually related to each other by a Fourier transformation,͉x ͘=1ͱ␲͵−ϱϱe −2ixp ͉p ͘dp ,͑26͒517S.L.Braunstein and P .van Loock:Quantum information with continuous variablesRev.Mod.Phys.,Vol.77,No.2,April 2005͉p͘=1ͱ͵−ϱϱe+2ixp͉x͘dx.͑27͒Despite being unphysical and not square integrable,the quadrature eigenstates can be very useful in calculations involving the wave functions␺͑x͒=͗x͉␺͘,etc.,and inidealized quantum communication protocols based on continuous variables.For instance,a vacuum state inﬁ-nitely squeezed in position may be expressed by a zero-position eigenstate͉x=0͘=͉͐p͘dp/ͱ␲.The physical,ﬁ-nitely squeezed states are characterized by the quadrature probability distributions͉␺͑x͉͒2,etc.,ofwhich the widths correspond to the quadrature uncer-tainties.B.Phase-space representationsThe Wigner function is particularly suitable as a “quantum phase-space distribution”for describing the effects on the quadrature observables that may arise from quantum theory and classical statistics.It behaves partly as a classical probability distribution,thus en-abling us to calculate measurable quantities such as mean values and variances of the quadratures in a classical-like fashion.On the other hand,in contrast to a classical probability distribution,the Wigner function can become negative.The Wigner function was originally proposed by Wigner in his1932paper“On the quantum correction for thermodynamic equilibrium”͑Wigner,1932͒.There, he gave an expression for the Wigner function in terms of the position basis which reads͑with x and p being a dimensionless pair of quadratures in our units withប=1/2as introduced in the previous section;Wigner, 1932͒W͑x,p͒=2␲͵dye+4iyp͗x−y͉␳ˆ͉x+y͘.͑28͒Here and throughout,unless otherwise speciﬁed,the in-tegration will be over the entire space of the integration variable͑i.e.,here the integration goes from−ϱtoϱ͒. We gave Wigner’s original formula for only one mode or one particle͓Wigner’s͑1932͒original equation was in N-particle form͔because it simpliﬁes the understanding of the concept behind the Wigner function approach. The extension to N modes is straightforward.Why does W͑x,p͒resemble a classical-like probability distribution?The most important attributes that explain this are the proper normalization,͵W͑␣͒d2␣=1,͑29͒the property of yielding the correct marginal distribu-tions,͵W͑x,p͒dx=͗p͉␳ˆ͉p͘,͵W͑x,p͒dp=͗x͉␳ˆ͉x͘,͑30͒and the equivalence to a probability distribution in clas-sical averaging when mean values of a certain class of operators Aˆin a quantum state␳ˆare to be calculated,͗Aˆ͘=Tr͑␳ˆAˆ͒=͵W͑␣͒A͑␣͒d2␣,͑31͒with a function A͑␣͒related to the operator Aˆ.The measure of integration is in our case d2␣=d͑Re␣͒d͑Im␣͒=dxdp with W͑␣=x+ip͒ϵW͑x,p͒,and we shall use d2␣and dxdp interchangeably.The opera-tor Aˆrepresents a particular class of functions of aˆand aˆ†or xˆand pˆ.The marginal distribution for p,͗p͉␳ˆ͉p͘,is obtained by changing the integration variables͑x−y =u,x+y=v͒and using Eq.͑26͒,that for x,͗x͉␳ˆ͉x͘,by using͐exp͑+4iyp͒dp=͑␲/2͒␦͑y͒.The normalization of the Wigner function then follows from Tr͑␳ˆ͒=1.For any symmetrized operator͑Leonhardt,1997͒,the so-called Weyl correspondence͑Weyl,1950͒,Tr͓␳ˆS͑xˆn pˆm͔͒=͵W͑x,p͒x n p m dxdp,͑32͒provides a rule for calculating quantum-mechanical ex-pectation values in a classical-like fashion according to Eq.͑31͒.Here,S͑xˆn pˆm͒indicates symmetrization.For example,S͑xˆ2pˆ͒=͑xˆ2pˆ+xˆpˆxˆ+pˆxˆ2͒/3corresponds to x2p ͑Leonhardt,1997͒.Such a classical-like formulation of quantum optics in terms of quasiprobability distributions is not unique.In fact,there is a whole family of distributions P͑␣,s͒of which each member corresponds to a particular value of a real parameter s,P͑␣,s͒=1␲2͵␹͑␤,s͒exp͑i␤␣*+i␤*␣͒d2␤,͑33͒with the s-parametrized characteristic functions ␹͑␤,s͒=Tr͓␳ˆexp͑−i␤aˆ†−i␤*aˆ͔͒exp͑s͉␤͉2/2͒.͑34͒The mean values of operators normally and antinor-mally ordered in aˆand aˆ†may be calculated via the so-called P function͑s=1͒and Q function͑s=−1͒,re-spectively.The Wigner function͑s=0͒and its character-istic function␹͑␤,0͒are perfectly suited to provide ex-pectation values of quantities symmetric in aˆand aˆ†such as the quadratures.Hence the Wigner function,though not always positive deﬁnite,appears to be a good com-promise in describing quantum states in terms of quan-tum phase-space variables such as single-mode quadra-tures.We may formulate various quantum states relevant to continuous-variable quantum communica-tion by means of the Wigner representation.These par-ticular quantum states exhibit extremely nonclassical features such as entanglement and nonlocality.Yet their Wigner functions are positive deﬁnite,and thus belong to the class of Gaussian states.518S.L.Braunstein and P.van Loock:Quantum information with continuous variables Rev.Mod.Phys.,Vol.77,No.2,April2005。

非线性光学 (Nonlinear Optics)

Robert W. Boyd
• Yujie Ding: B.S., Electronic Sciences, Jilin University; M.S., Electrical Engineering, Purdue Univ.; Ph.D.,Electrical Engineering, Johns Hopkins Univ.; now professor at Lehigh Univ..
• 其中虚线代表的是偏振
和电场之间的线性关系，而实线代表非线性关系。
（a）在小电场情况下，偏振与电场的时间变化接近。（b）电场强度加大后，偏振响应产生了非对称性，在负电场情况下具有较大的偏离。 • 以上这种失真的输出在电路理论中可以解释为高阶简谐成份的出现。
二、光学非线性的物理起源
Non-resonant nonlinearities 非共振非线性 •由，令，有。 • 即在不为零时，频率为ω的入射光场在介质中产生了频率为2ω的出射光场。
• 将一个电子束缚到一个原子中的电场幅度在1010-1011 V m-1左右，在光电场幅度与该数值接近时非线性效应开始凸显。
•由
，此时光强需要达到1019 W m-2，可以由高功率激光来实现。
• 实际上并不需要上述的高光强，因为大量原子的微弱非线性效应可以叠加在一起产生可观的宏观非线性效应 – 需要相位同步即“phase matching”条件。
原子跃迁的光谱线型函数
二、光学非线性的物理起源
Resonant nonlinearities 共振非线性 • 同样可以得到介质的单位时间内添加到光束中的受激发射光子数目：
• 此时单位时间内减少的光子数目为
，即净吸收速率。
• 随着光束在介质中的传播，其强度逐渐减小：定义z处的光强为I(z)，dz内光强的变化为dI ，此时有。 • 由于光束强度定义为单位时间在单位面积上通过的能量（W m-2），有，即。

量子信息和量子纠缠理论

0, E ( ) H 2 [ ()] [1 ()]log2 (m ห้องสมุดไป่ตู้ 1), log2 (m 1) ( m) log m, 2 m2
verschrä nkt entangled
1 01 10 2

Alice
Bob
Entanglement enhanced
1 qubit
2 bit
1 qubit
Alice
Bob
Entanglement enhanced
Alice
Bob
1 01 10 2

Entanglement enhanced
Separable! Separable！
Separability of mixed states: no general criteria a) Peres (PPT) criterion:
Peres PRL 77, 1413 (1996)
2x2, 2x3:
PPT
Separable
Horodeckis, Phys. Lett. A 223,1 (1996)
Copier
Alice
Eve
Bob
No-Cloning Copier:
Quantum information is a new kind of information
eavesdropping on quantum information
?
Copier
Alice
Eve
Bob
Detected
eavesdropping on quantum information
14 24 34 44

Absolutely Maximally Entangled States

Absolutely Maximally Entangled States:Existence and ApplicationsWolfram Helwig and Wei CuiCenter for Quantum Information and Quantum Control (CQIQC),Department of Physics,University of Toronto,Toronto,Ontario,M5S 1A7,CanadaJune 12,2013AbstractWe investigate absolutely maximally entangled (AME)states,which are multipartite quantum states that are maximally entangled with re-spect to any possible bipartition.These strong entanglement properties make them a powerful resource for a variety of quantum information pro-tocols.In this paper,we show the existence of AME states for any number of parties,given that the dimension of the involved systems is chosen ap-propriately.We prove the equivalence of AME states shared between an even number of parties and pure state threshold quantum secret sharing (QSS)schemes,and prove necessary and suﬃcient entanglement proper-ties for a wider class of ramp QSS schemes.We further show how AME states can be used as a valuable resource for open-destination teleporta-tion protocols and to what extend entanglement swapping generalizes to AME states.1IntroductionEntanglement has been a hot topic since the beginning of quantum mechanics and fueled a lot of discussions,among them most notable the Einstein-Podolsky-Rosen (EPR)paradox [1],which ﬁnally led Bell to come up with a method of actually measuring entanglement [2].It was not until the advent of quantum information,however,that entanglement was recognized as a useful resource.Almost all applications in quantum information make either explicit or implicit use of entanglement,which makes it crucial to gain as much insight as possible.[3]While the entanglement of bipartite states is already very well understood[4,5,6],the road to its generalization to more than two parties is paved with many obstacles.Therefore we often have to restrict ourselves to special cases when analyzing multipartite entanglement.A prominent choice are states that extremize the entanglement for a certain measure of entanglement.In this paper we want to do that by focusing on absolutely maximally entangled (AME)states,which are deﬁned as states that are maximally entangled for any possible bipartition.[7,8,9]1a r X i v :1306.2536v 1 [q u a n t -p h ] 11 J u n 2013Deﬁnition1.An absolutely maximally entangled state is a pure state,shared among n parties P={1,...,n},each having a system of dimension d.Hence |Φ ∈H1⊗···⊗H n,where H i∼=C d,with the following equivalent properties: (i)|Φ is maximally entangled for any possible bipartition.This means thatfor any bipartition of P into disjoint sets A and B with A∪B=P and, without loss of generality,m=|B|≤|A|=n−m,the state|Φ can be written in the form|Φ =1√d mk∈Z md|k1 B1···|k m B m|φ(k) A,(1)with φ(k)|φ(k ) =δkk .(ii)The reduced density matrix of every subset of parties A⊂P with|A|= n2 is totally mixed,ρA=d− n2 1d n2.(iii)The reduced density matrix of every subset of parties A⊂P with|A|≤n2 is totally mixed.(iv)The von Neumann entropy of every subset of parties A⊂P with|A|= n2 is maximal,S(A)= n2 log d.(v)The von Neumann entropy of every subset of parties A⊂P with|A|≤n2 is maximal,S(A)=|A|log d.These are all necessary and suﬃcient condition for a state to be absolutely max-imally entangled.We denote such a state as an AME(n,d)state.The simplest examples of AME states occur for low dimensional systems shared among few parties.Starting with qubits,the most obvious one is an EPR pair,which is maximally entangled for its only possible bipartition.For three qubits shared among three parties,we can recognize the GHZ state as an AME state.It is maximally entangled,with1ebit of entanglement with respect to every bipartition.For four qubits,there is no obvious candidate,and in fact it has been shown that for four qubits no AME state exists[9].We can stillﬁnd an absolutely maximally entangled states for four parties,however,by increasing the dimensions of the involved systems.An AME(4,3)state for four qutrits shared among four parties exists,and it is given by[7]|Φ =1√92i,j=0|i |j |i+j |i+2j .(2)This is theﬁrst indicator that the search for AME states gets more promising as we increase the dimensions of the systems.Completing the characterization of AME states for qubits,it is known that AME states exist for5and6qubits.Explicit forms for them are given in Ref.[7], and it turns out that they are closely related to theﬁve-qubit error correction code.For7qubits,it is still not known if an AME state exists,whereas for≥8 qubits,it has been shown that no AME states can exist[9,10].In Ref.[7],we showed how AME states can be used for parallel teleportation protocols.In these protocols,the parties are divided into a sets of senders2and receivers,respectively.One of the two sets is given the ability to perform joint quantum operations,while players in the other set can only perform local quantum operations.Under these conditions,a parallel teleportation of multiple quantum states is possible if the set that performs joint quantum operations is larger than the other set.A closer look at these teleportation scenarios then led to the observation that any AME state shared by an even number of parties can be used to construct a threshold quantum secret sharing(QSS)scheme[11, 12,13].The opposite direction was also shown,with one additional condition imposed on the QSS scheme,namely that the shared state that encodes the secret is already an AME state.In this paper,we will give an information-information theoretic proof of this equivalence of AME states and threshold QSS scheme,which shows that the additional condition is not required.We will rather see that it is satisﬁed for all threshold QSS schemes.We will further give a recipe of how to construct AME states from classical codes that satisfy the Singleton bound[14].This construc-tion can be used to produce AME states for a wide class of parameters,and it even proves that AME states exist for any number of parties for appropriate system dimension.A result that could also be deduced from the equivalence of AME states and QSS schemes and a known construction for threshold QSS schemes[11].We will then show more applications for AME states.Theﬁrst be-ing the construction of a wider class of QSS schemes,the ramp QSS schemes,of which threshold QSS schemes are a special case.The next one is the utilization of AME states as resources for open-destination teleportation protocols[15]. Finally,we investigate to what extend entanglement can be swapped between two AME states.This paper is structured as follows.In Section2,we show how AME states can be constructed from classical codes,which also also shows the existence of AME states for any number of parties.In Section3,we establish an equiv-alence between even party AME states and threshold QSS schemes,using an information theoretic approach to QSS schemes.Section4shows how to share multiple secrets using AME states.In Section5,we show that AME states can be used for open-destination teleportation.After that,swapping of AME states is investigated in Section6.2Constructing AME States from Classical MDS CodesThere is a subclass of AME(n,d)states that can be constructed from optimal classical error correction codes.A classical code C consists of M codewords of length n over an alphabetΣof size d.For our purposes,the alphabet isgoing to beΣ=Z d and thus C⊂Z nd .The Hamming distance between twocodewords is deﬁned as the number of positions in which they diﬀer,and the minimal distanceδof the code C as the minimal Hamming distance between any two codewords.For a given length n and minimal distanceδ,the number of codewords M in the code is bounded by the Singleton bound[14,16]M≤d n−δ+1.(3) Codes that satisfy the Singleton bound are referred to as maximum-distance separable(MDS)codes.They can be used to construct AME states:3Theorem2(a).From a classical MDS code C⊂Z2md of length2m and minimaldistanceδ=m+1over an alphabet Z d,an AME(2m,d)state can be constructed as|AME =1√d mc∈C|c (4)=1√d mc∈C|c1 1···|c m m|c m+1 m+1···|c2m 2m.(5)Proof.The code C satisﬁes the Singleton bound,which means the sum contains a total of M=d2m−δ+1=d m terms.Furthermore,any two of these terms diﬀer in at least one of theﬁrst m kets because the code has minimal distance δ=m+1.Hence the sum contains each possible combination of theﬁrst m basis kets exactly once.Moreover,for any two diﬀerent terms,the last m kets must also diﬀer in at least one ket and are thus orthogonal.This means the state has the form of Equation(1)with respect to the bipartition into theﬁrst m and last m parties.The same argument works for any other bipartition into two sets of size m,hence the state is absolutely maximally entangled.An analogous argument shows that a similar construction for an odd number of parties results in an AME state.Theorem2(b).From a classical MDS code C⊂Z2m+1d of length2m+1andminimal distanceδ=m+2over an alphabet Z d,an AME(2m+1,d)state can be constructed as|AME =1√d mc∈C|c (6)=1√d mc∈C|c1 1···|c m+1 m+1|c m+2 m+2···|c2m 2m+1.(7)Proof.The code contains M=d m terms.Each of the terms diﬀer in at least one of theﬁrst m+1and last m terms.Thus,with the same argument as above, this is an AME state.Trivial states of that form are d-dimensional EPR states,which are repre-sented by the code with codewords00,11,...,(d−1)(d−1).This code has n=2,δ=2,M=d1.For n=3,we canﬁnd the GHZ states for arbitrary dimensions, which can be constructed from the code000,111,...,(d−1)(d−1)(d−1),which hasδ=3and M=d1.As already mentioned in the introduction,for n=4 no AME state exists for d=2,however for d=3the AME(4,3)state given in Equation(2)can also be constructed from an MDS code,the[4,2,3]3ternary Hamming code.A wide class of MDS codes is given by the Reed-Solomon codes and its generalizations[17,16,18],which give MDS codes for n=d−1,n=d,and n=d+1,for d=p x being a positive power of a prime number p.From the Reed-Solomon codes,MDS codes can also be constructed for n<d−1[14].This shows that AME states exist for any number of parties if the system dimensions are chosen right.At this point we would like to mention that after posting a preliminary ver-sion of our last paper on this subject[7],it has been brought to our attention by4Gerardo Adesso that the results of this section have already been previously dis-covered by Ashish Thapliyal and coworkers,and were presented at a conference in 2003[19],but remained unpublished.3Equivalence of AME states and QSS schemes In Ref.[7],we showed that AME(2m,d )states,i.e.,AME states shared between an even number of parties,are equivalent to pure state threshold quantum secret sharing (QSS)schemes that have AME states as basis states and share and secret dimension equal to d .Here we will give an information-theoretic proof of this equivalence,which shows that the requirement that the basis states of the QSS scheme are AME states is redundant,as it follows from this proof that these states are always absolutely maximally entangled.Before stating the theorem and the proof,we give a short motivation why AME states and QSS schemes are related.Consider an AME(2m,d )state shared among an even number of parties.If we take any bipartition into two sets of parties A and B ,each of size m ,a d m dimensional state can be teleported from one set to the other due to the maximal entanglement between A and B .Moreover,we have shown in Ref.[7],that the teleportation can be performed in such a way that each party in the sending set B performs a local teleportation operation on their qudit,while the parties in the receiving set A perform a joint quantum operation to recover all m teleported qudits.This is depicted in Figure 1for the case of m =4.This also works if only one party in B ,which we call the dealer D ,performs the teleportation operation,while the others do nothing.Then the teleported d -dimensional state can still be recovered by the players in set A .Furthermore,this also works for any other bipartition into sets A and B of size m ,with D ∈B ,without changing the teleportation operation D has to perform,but now the parties in A can recover the teleported state (see Figure 2).This means that any set with m parties can recover the state.Moreover,the no-cloning theorem guarantees that the complement of a set that can recover the state has no information about the state.Hence all sets with less than m parties cannot gain any information about the state.This,however,are exactly the requirements for a threshold QSS scheme,therefore we have constructed a ((m,2m −1))threshold QSS scheme from the AME(2m,d )state.To formally show this,and moreover that it also works in the opposite direction,meaning that a ((m,2m −1))threshold QSS scheme is always related to an AME(2m,d )state,we will use the information theoretic description of QSS schemes as introduced in Ref.[13].Let us quickly review the framework for a pure state ((m,2m −1))threshold QSS scheme [11].A secret S is distributed among the players P ={1,...,2m −1}such that any set A ⊆P with |A |≥m can recover the secret,while any set B ⊂P with |B |<m cannot gain any information about the secret.We further only consider the case where the dimension d of the secret is the same as the dimension of each player’s share.The secret is assumed to lie in the Hilbert space H S ∼=C d ,and the share of party i in H i ∼=C d .The encoding is described by an isometryU S :H S →H 1⊗···⊗H 2m −1.(8)The secret S is chosen randomly and thus is described by ρS =1/di |i i |.We5Figure 1:(Color online)Parties in B (green)perform local teleportation op-erations,parties in A (red)can recover teleported states by performing a joint quantumoperationFigure 2:(Color online)After D (blue)performs her teleportation operation,any set of m parties (red),A ,A ,A etc.,can recover the teleported state.Any set of parties with m −1or less parties (any set consisting only of green parties)cannot gain any information about the teleported state.consider its puriﬁcation by introducing a reference system R such that |RS =1/√d i |i |i ∈H R ⊗H S .Let ρRA denote the combined state of the reference system and a set of players A ⊆P after U S has been applied to the secret.Then the players A can recover the secret,if there exists a completely positive map T A :H A →H S such that [13,20]1R ⊗T A (ρRA )=|RS .(9)This can be stated in terms of the mutual informationI (X :Y )=S (X )+S (Y )−S (X,Y )(10)as follows:Deﬁnition 3.An isometry U S :H S →H 1⊗···⊗H 2m −1creates a ((m,2m −1))threshold QSS scheme if and only if,after applying to the system S of the pu-riﬁcation |RS ,the mutual information between R and an authorized (unautho-rized)set of players A (B )satisﬁesI (R :A )=I (R :S )=2S (S )if |A |≥m (11)I (R :B )=0if |B |<m.(12)6Here S is the von Neumann entropy,and because of S(i)≥S(S)[13],we haveS(S)=S(R)=S(i)=log d.(13) From Equations(10)to(12)it immediately follows thatS(R,A)=S(A)−S(R)if|A|≥m(14)S(R,B)=S(B)+S(R)if|B|<m.(15)Theorem4.For a state|Φ the following two properties are equivalent:(i)|Φ is an AME(2m,d)state.(ii)|Φ is the puriﬁcation of a((m,2m−1))threshold QSS scheme,whose share and secret dimensions are d.Proof.(i)→(ii):We need to show that for an AME(2m,d)state Equations(11) and(12)are satisﬁed,where R can be any of the2m party.This follows directly from the deﬁnition of the mutual information,Equation(10),and Deﬁntion1 (v).(ii)→(i):Consider an unauthorized set of players B,with|B|=m−1. Then the set is B∪i is authorized for any additional player i/∈B,and from Equation(14)we haveS(B,i,R)=S(B,i)−S(R)(16)On the other hand,using the Araki-Lieb inequality[21]S(X,Y)≥S(X)−S(Y) and Equation(15)givesS(B,i,R)≥S(B,R)−S(i)=S(B)+S(R)−S(i).(17)Combining the last two equations and using S(S)=S(R)=S(i)showsS(B,i)≥S(B)+S(i),(18)where equallity must hold due to the subadditivity of the entropy S(X,Y)≤S(X)+S(Y).This means that the entropy increases maximally when adding one player’s share to m−1shares.The strong subadditivity of the entropy[21]S(X,Y)−S(Y)≥S(X,Y,Z)−S(Y,Z)(19)states that adding one system X to a system Y increases the entropy at least by as much as adding the system X to a larger system Y∪Z that contains Y. So in our case,adding one share to less than m−1shares increases the entropy by at least S(i),and since this is the maximum,it increases the entropy exactly by S(i).Hence,starting out with a set of no shares,and repeatedly adding one share to the set until the set contains any m shares and is authorized,shows that any set of m shares has entropy mS(i).This shows that the entropy is maximal for any subset of m parties and thus|Φ is an AME(2m,d)state.Corollary5.The encoded state U S|S of a speciﬁc secret|S with a((m,2m−1))threshold QSS protocol with share and secret dimension d is an AME(2m−1,d)state.74Sharing multiple secretsIn the previous section,we outlined how an AME state can be used to construct a QSS scheme.The role of the dealer is assigned to one of the parties and he performs a teleportation operation on his qudit,which encodes the teleported qudit onto the qudits of the remaining parties such that the criteria for a QSS scheme are met.While Theorem 4shows the equivalence of AME states and QSS schemes,the actual protocol for the encoding and decoding operations has been presented in Ref.[7].Note that in the described scenario,the role of the dealer can be assigned to any player.Thus one may ask,what happens if more than one of the players assumes the role of the dealer.The answer is that,given an AME(2m,d )state,up to m players are able to independently encode one qudit each onto the qudits of the remaining players in such a way that results in a QSS scheme with a more general access structure.For a secret sharing scheme with a general access structure,each set of players falls into one of three categories [22,23].1.Authorized :A set of players is authorized,if it can recover the secret2.Forbidden :A set of players is called a forbidden set,if the players cannot gain any information about the encoded secret3.Intermediate :A set of players is classiﬁed as an intermediate set,if they cannot recover set secret,but may be able to gain part of the information.This means that the reduced density matrix of that set of players depends on the encoded secret,but not enough as to recover the secret.A special kind of access structure is a (m,L,n )ramp secret sharing scheme[24].Here n is the total number of players,m is the number of players needed to recover the secret,and L is the number of shares that have to be removed from a minimal authorized set to destroy all information about the secret.In terms of the above deﬁned set categories that means that any set of m or more players is authorized,any set of m −L or less players is forbidden,and any set consisting of more than m −L ,but less than m players is an intermediate set.This is the access structure we get from an AME(2m,d )state if more than one party assumes the role of the dealer.Theorem 6.Given an AME(2m,d )state,a QSS scheme with secret dimension d L and a (m,L,2m −L )ramp access structure can be constructed for all 1≤L ≤m .Proof.The encoding of the secret is done by assigning the role of dealer to L of the 2m players.For simplicity we choose them to be the ﬁrst L players.Each of them performs a Bell measurement on their respective qudit of the AME state and one qudit of the secret.The Bell measurement is described by the general d -dim Bell states |Ψkl and the unitaries U kl that transform among them [25]|Ψqp =1√d j e 2πijq/d |j |j +p (20)U qp = j e 2πijq/d |j j +p |,(21)8where the kets are understood to be mod d .For a secret |s and outcomes (q 1,p 1)...(q L ,p L )for the Bell measurement of the dealers,the initial AME(2m,d )state is transformed to|ΦS =1√ k ∈Z m ds qp ,k 1···k L |k L +1 B 1···|k m B m −L |φ(k ) A .(22)Heres qp ,k 1···k L = k 1···k L |U †q 1p 1⊗···⊗U †q L p L |s ,(23)and the partition of the remaining 2m −L parties into two sets A and B of size m and m −L ,respectively,is arbitrary.After obtaining their measurement outcomes,the dealers broadcast their results to all of the remaining players.This concludes the encoding process.To show that any set of m or more players is authorized,it suﬃces to show that set A in Equation22can recover the secret.They can do so by applying the unitary operationU =(U q 1p 1⊗···⊗U q L p L ⊗1)V(24)withV =k ∈Z m d |k 1 ···|k m φ(k )|,(25)to their system.This changes the state toU |ΦS =1√d m −L (k L +1,...,k m )∈Z m −L d |k L +1 B 1···|k m B m −L |s A |k L +1 A L +1···|k m A m (26)where A ={A 1,...,A L }.Thus the players in set A have the secret in their possession.It immediately follows from the no-cloning theorem that B ,and thus any set of size m −L or less,cannot have any information about the secret since all information is located in the complement set.Alternatively,this also follows from the observation that the reduced density matrix of B is always completely mixed,independent of the secret.The last thing left to show is that all sets with more than m −L but fewer than m players are indeed intermediate sets.To see that,consider the case L =1,where a set C of m −1players is not authorized to recover the secret.If one more player in the complement of C assumes the role of the dealer,the scheme is changes to L =2.This operation does not change the fact that C cannot recover the ﬁrst secret,and thus it is still not authorized for L =2.This argument can be continued to any other 1<L ≤m by adding more dealers.Hence a set of m −1(or fewer)players is not authorized to recover the secret for all value of 1≤L ≤m .That a set of more than m −L players is not forbidden follows from the fact that information cannot be lost and thus the complement of a forbidden set has to be authorized.However,we just argued that the complement of a set of more than m −L players is not authorized (since it consists of less than m players).Hence any set with more than m −L and fewer than m players is an intermediate set.9A closer look at the proof shows us that it actually is not absolutely necessary for the initial state to be maximally entangled with respect to any bipartition,but only for bipartitions for which all dealers are in the same set.In fact,we can generalize the proof of Theorem 4to the case of ramp QSS to show that this is a necessary and suﬃcient condition for the construction of (m,L,2m −L )ramp QSS schemes.Theorem 7.For a state |Φ ∈H P ⊗H R ,shared between 2m −L players P ,each holding a qudit,and L reference qudits,the following two properties are equivalent:(i)|Φ is maximally entangled for any bipartition for which the L referencequdits are in the same set.(ii)|Φ is the puriﬁcation of a (m,L,2m −L )ramp QSS schemes.The encodedsecret of the ramp QSS scheme has dimension d L ,and each share has dimension d .The proof is a straightforward generalization of the proof of Theorem 4and is provided in Appendix A.5Open-destination teleportationGiven a state with such high amount of entanglement as the AME state has,one cannot help thinking about ways of using these resources for teleportation protocols.In Ref.[7]we already showed how AME states can be used for two diﬀerent teleportation scenarios that require either sending or receiving parties to perform joint quantum operations,while the other end may only use local quantum operations.Another teleportation scenario that uses genuine multipartite entanglement,and has already been demonstrated experimentally [15],is open-destination tele-portation.In this scenario,a genuinely multipartite entangled state is shared between n parties,each in the possession of one qudit.One of the parties,the dealer,performs a teleportation operation on her qudit and an ancillary qudit |Φ .After this teleportation operation,the ﬁnal destination of |Φ is still un-decided,thus open-destination teleportation.The destination is decided upon in the next step,where a subset A of the remaining parties P performs a joint quantum operation on their qudits such that a player in P \A ends up with the state |Φ –up to local operations that depend on measurement outcomes of the dealer and parties A .Here we want to show that open-destination teleportation can also be performed with AME states.Assume that an AME(n,d )state has been distributed among n parties.One of the n parties is assigned the role of the dealer.She performs a Bell measurement on her qudit and the secret |S = a i |i .This transforms the state to|S |Φ →|ΦS =1√d m (k,i )∈Z m da pq,i |k 1 B 1···|k m −1 B m −1|φ(k,i ) A ,(27)where pq labels the outcome of the Bell measurement and has to be made public.The remaining n −1parties that share the resulting state have been divided10into two sets A and B of size n/2 and m−1= n/2 −1,respectively.Now,after the teleportation operation has been completed,the parties in set Amay choose one party B i∈B as theﬁnal destination for the state|S .Then, after performing the joint unitary operation of Equation(25)followed by a Bellmeasurement on qudits A i and A m with outcome rs,the party B i ends up withthe state|ΦB i =U†rs U†pq|S ,which can be easily transformed to|S if themeasurement results pq and rs are known.Note that with the parallel teleportation protocol introduced in Ref.[7],also one of the parties in A can be chosen to receive the state|S .Thus,after the dealer’s teleportation operation is completed,any set of size greater or equal n/2 can choose any of the remaining n−1parties as theﬁnal destination of the teleportation.116Swapping of AME statesEntanglement swapping [26]is a very useful tool for the application of entan-glement in communication.By making a Bell measurement on Bob’s side,two entangled states shared between Alice and Bob,and Bob and Charlie,respec-tively,can be transformed into an entangled state shared by Alice and Charlie.Employing this procedure in quantum repeaters [27]allows entangled states to be used for long distance communications.In this section,we show to what ex-tent a generalization of the entanglement swapping protocol can be constructed to allow swapping of entanglement between absolutely maximally entangled states shared between diﬀerent parties.Assume that parties {1,2,...,2n }share an AME(2n,d )state,|Φ 1,...,2n = |i 1···i n 1,...,n |φ(i 1,...,i n ) n +1,...,2n (28)= |i 1···i n 1,...,n U |i 1···i n n +1,...,2n ,(29)where U is a unitary transformation with U |i 1···i n =|φ(i 1,...,i n ) .Suppose parties {n +1,...,3n }also share an AME(2n,d )state|Φ n +1,...,3n = |i 1···i n n +1,...,2n U |i 1···i n 2n +1,...,3n .(30)Now each of the parties {n +1,...,2n }performs a Bell measurement on their qudits from both AME states.Without loss of generality,we can assume the measurement result is (q,p )=(0,0)(see Equation (20)for the notation),since other measurement outcomes produce the same state up to local transforma-tions.Then the state shared by the parties {1,...,n,2n +1,...,3n }becomes|Φ 1,...,n,2n +1,...,3n = |i 1···i n 1,...,n U 2|i 1···i n 2n +1,...,3n (31)Consecutive applications of the above procedure gives the following lemma:Lemma 8.Suppose each group of parties {1,...,2n },{n +1,...,3n },···,{mn +1,...,(m +1)n }shares an AME(2n,d )state,|Φ = |i 1···i n U |i 1···i n .(32)Then,if each of the parties {n +1,n +2,...,mn }performs a Bell measurement on their two qudits,the resulting state shared by the parties {1,...,n,mn +1,...,(m +1)n }is locally equivalent to|Φ 1,...,n,mn +1,...,(m +1)n = |i 1···i n 1,...,n U m |i 1···i n mn +1,...,(m +1)n (33)Proof by induction.The case for m =2is demonstrated in the above discussion already.If the lemma holds for m ,for m +1the two remaining states,after the parties {n +1,n +2,...,mn }performed their Bell measurements,are |Φ 1,...,n,mn +1,...,(m +1)n = |i 1···i n 1,...,n U m |i 1···i n mn +1,...,(m +1)n (34)12。

总结欧拉36片拼图有一个量子解决方案

总结欧拉36片拼图有一个量子解决方案大名鼎鼎的瑞士数学家欧拉（Leonhard Euler）一直是数学领域关键性人物，他提出的著名的“36 名军官之谜”在过去243年，一直无解。

如今，一个令人惊讶的新解决方案提供了一种编码量子信息的新方法，有效的“解决了”该问题。

1. “不可能”六人军团问题早在1779年，欧拉提出了一个很有名的不可解难题：六个军团各有六个不同军衔的军官，这36名军官能否被安排在一个6乘6的方阵中，使任何一行或一列的军官的军衔或所在军团都不重复？当问题变成有5个军衔和5个军团，或7个军衔和7个军团时，这个谜题就很容易解决了。

但6个军衔和6个军团36名军官的情况却无法寻求到合适的解决方案。

在绞尽脑汁找解决方案未果后，欧拉得出结论："这样的安排是不可能的，尽管我们无法对此给出严格的证明"。

一个多世纪后，法国数学家Gaston Tarry证明，确实没有办法将欧拉提出的36名军官安排在一个6乘6的正方形中而不重复。

2.计算机辅助，仍然无解1960年，数学家们用计算机来证明[1]，任何数量的军团和军衔都存在相应的解决方案，但奇怪的是，说巧不巧唯独6个军团的情况除外。

两千多年来，类似的谜题一直让人们着迷。

世界各地的文化都制作了 "魔法方块(Magic squares)"，即构建每一行和每一列的数字相加之和相同的方块，以及每一行和每一列都出现一次的充满符号的 "拉丁方块"。

以这一灵感设计的文化广场已被用于艺术和城市规划，但也只是为了有趣和引人思考。

一个较为流行的拉丁方格（数独）的子方格也缺乏重复的符号。

欧拉的36个军官谜题要求一个 "正交拉丁方块"，其中两组属性，如军衔和军团，都同时满足拉丁方块的规则。

3.量子版本挑战“不可能”尽管200多年前的欧拉认为不存在这种6乘6的正方形。

如今，游戏规则发生了变化。

在近期发布并提交给《物理评论快报》的一篇论文中[2]，印度和波兰合作的量子物理学家证明，有可能以符合欧拉标准的方式安排36名军官。

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a r X i v :q u a n t -p h /0402059v 1 8 F e b 2004Nonmaximally-entangled-state quantum photolithographyYan-Hui Wang and Le-Man Kuang ∗†Department of Physics,Human Normal University,Changsha 410081,ChinaMany previous works on quantum photolithography are based on maximally-entangled states (MES).In this paper,we generalize the MES quantum photolithography to the case where two light beams share a N -photon nonmaximally-entangled state.we investigate the correlations between quantum entanglement and quantum photolithography.It is shown that for nonlocal entanglement between the two light beams the amplitude of the deposition rate can be changed through varying the degree of entanglement described by an entanglement angle while the resolution remains unchanged,and found that for local entanglement between the two light beams the eﬀective Rayleigh resolution of quantum photolithography can be resonantly enhanced.PACS number(s):42.50.Dv,42.25.Hz,85.40.HpI.INTRODUCTIONRecently,much attention has been paid to quantum photolithography [1,2,3,4]due to the possibility of beating the classical Rayleigh diﬀraction limit through using on quantum entanglement between two used light beams.Optical lithography is a widely used printing method which has been the primary tool of the semi-conductor industry for transferring circuit images onto substrates to produce smaller and smaller processors.In this process,light is used to etch a substrate and the exposed or unexposed areas on the substrate deﬁne pat-terns.The resolution of images transferred by using clas-sical light beams is restricted to the Rayleigh diﬀraction limit λ/4,λbeing the wavelength of the light,hence one can achieve a resolution only comparable to the wave length of the light used in classical optical lithography [5,6,7].In Ref.[3],Agedi and coworkers introduced a procedure called quantum lithography in one dimen-sion that predicts an increase in resolution beyond the diﬀraction limit due to quantum entanglement between two light beams,and demonstrate a quantum lithogra-phy method to improve the resolution by a factor N in contrast to classical one,using N -photon maximal entan-gled state (MES)[8].The Maryland group [2]completed a proof-of-principle experimental demonstration of quan-tum lithography by using two-photon entangled state generated via a specially designed spontaneous paramet-ric down-conversion.The increase in resolution makes quantum lithography a potentially useful tool to pro-duce smaller computer chips in nanotechnology.Then the one-dimensional quantum lithography method was generalized to the two-dimensional case [9]and entan-gled binomial states [10].Since the number of elements writable on a surface scales inverse quadratically with the minimum feature dimension,this improvement is an important advance.The purpose of the present paper isN !,(2)with the superposition mode operator ˆe =(ˆa +ˆb )/√2.If a substrate is exposedfor a time t to the light source,the exposure function P (φ)=∆N t gives an exposure pattern.Before going into the NMES lithography,let us brieﬂy review the MES quantum lithography presented in Ref [3].Consider two counterpropagating light beams a and2 b cross each other at the surface of a photosensitive sub-strate.They have a relative phase diﬀerenceφ=kx/2with the optical wave number k=2π/λand x is thelateral dimension on the substrate to describe the po-sition where the two beams meet.For the N-photonmaximally-entangled number state of the two light beams|ψN ab=12(|N,0 ab+e iNφ|0,N ab),(3)from Eqs.(1)and(2)one can get the following depositionrate∆N(φ)=14N,(5)which increases the resolution by a factor N in contrastto the classical diﬀraction limit∆x=λ/4.Now we consider the quantum lithography with a N-photon NMES given by|ψN ab=cosγ|N,0 ab+e iNφsinγ|N,0 ab,(6)whereγmeasures the entanglement of the state,itchanges from0(no entanglement)toπ/4(maximal en-tanglement).For the N-photon NMES(6),from Eqs.(1)and(2)we obtain the following deposition rate∆N(γ,φ)=12N+1[2+sin(k+1)Nφ+sin(k−1)Nφ],(8)which leads to the following resolution∆x=12N+1[2+sin(2Nφ)],(10)which leads to the following resolution∆x=13 to the N-photon NMES(12),we have to calculate thematrix elements of the dosing operator∆Nm′Nm= ψNm|ˆδN|ψNm′ ,(13)which is given by∆Nm′Nm (γ,φ)=1C mNC m′N cos2γe i(m′−m)φ+sin2γe−i(m′−m)φe i(θm′−θm)+12NC m N{1+sin(2γ)cos[(N−2m)φ+θm]}(15)In particular,whenγ=π/2,the deposition rate(15) becomes∆Nm(π/2,φ)=12NNn=0C m N|C n|2{1+cos[(n−2m)φ+θm]sin(2γ)},(20)which indicates that the exposure pattern induced by the superposition state(17)is determined by the moduleof the superposition coeﬃcients|C n|,the entanglement angleγ,and the relative phaseθn.From Eq.(20)it is easy to understand the role of quantum entanglement in quantum photolithography.In fact,from Eq.(20)we can see that varying of the entan-glement angleγis equivalent to rescaling time parameter t or/and the module of the superposition coeﬃcient|C n| for the case of nonlocal entanglement where the entan-glement angleγis independent of the associated phase diﬀerenceφ.In particular,whenγ=π/4,Eq.(20)re-duces to the expression of exposure pattern of the MES case[8]P(φ)=tand a standard truncated Fourier seriesP (φ)=Qt +tNn =0(a n cos nφ+b n sin nφ),(22)where Q is the uniform background penalty exposure rateQ =Nn =0|C n |2,(23)and the expanding coeﬃcients are determined by themodule of the superposition coeﬃcients |C n |,the entan-glement angle γ,and the relative phase θn with the fol-lowing expressionsa n =|C n |2sin(2γ)cos θn ,b n =|C n |2sin(2γ)sin θn .(24)From Eqs.(22),(23),and(24)wecansee that quantum entanglement between two light beams dos not change the background penalty exposure rate but it con-trols the amplitudes of all Fourier components.It is well known that any suﬃciently well-behaved pe-riodic function can be written as an inﬁnite Fourier se-ries.However,when we create patterns with the pseudo-Fourier lithography method,we do not have access to ev-ery component of the Fourier expansion,since this would involve an inﬁnite number of photons .This means that we can only employ truncated Fourier series,and these can merely approximate arbitrary patterns.The Fourier expansion has the nice property that when a series is truncated at N ,the remaining terms still give the best Fourier expansion of the function up to N .In other words,the coeﬃcients of a truncated Fourier series are equal to the ﬁrst N coeﬃcients of a full Fourier series.As an example,in what follows we use the pseudo-Fourier method to simulate a pattern generated by the following test functionF (φ)=|sin φ|,(25)which can be expanded as a Fourier seriesF (φ)=2π∞n =1cos(2nφ)πt sin(2γ)(4n 2−1).(28)Substituting Eqs.(27)and (28)into Eq.(17),one canobtain the superposition state to realize the pattern of the test function given by Eq.(25).And from (27),(28)and (17)we can see that the superposition state consists of even-number-photon NMES like (12),and only the module of the superposition coeﬃcients aﬀect the depo-sition rate of quantum photolithography.In Figure 1we have simulated the test pattern (25)by using the super-position state given by Eq.(24)for N =2,6,and 12cases,respectively.From Figure 1we can see that the calculated patterns are in good agreement with the test pattern as shown by the solid curve,and the larger the value of N is,the better the eﬀect of the simulation.IV.CONCLUDING REMARKSIn conclusion we have generalized the N -photon MES quantum photolithography to the N -photon NMES case,and investigated the correlations between quantum en-tanglement and quantum photolithography.It has been found that quantum photolithography can be manipu-lated and controlled through varying quantum entangle-ment between two applied light beams.Especially,for the nonlocal entanglement case,we have showed that the amplitude of the deposition function increases with the degree of entanglement.Hence one can manipulate and control the amplitude of the 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